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1
Kamel, Aladin H. "A stability checking procedure for finite-difference schemes with boundary conditions in acoustic media." Bulletin of the Seismological Society of America 79, no. 5 (October 1, 1989): 1601–6. http://dx.doi.org/10.1785/bssa0790051601.
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Abstract The manner in which boundary conditions are approximated and introduced into finite-difference schemes has an important influence on the stability and accuracy of the results. The standard von Neumann stability condition applies only for points which are not in the vicinity of the boundaries. This stability condition does not take into consideration the effects caused by introducing the boundary conditions to the scheme. In this paper, we extend the von Neumann condition to include boundary conditions. The method is based on studying the time propagating matrix which governs the space-time behavior of the numerical grid. Examples of applying the procedure on schemes with different boundary conditions are given.
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Li, Yong Heng, and Xia Li. "Extension of Von Neumann Model of National Economic System." Applied Mechanics and Materials 55-57 (May 2011): 101–4. http://dx.doi.org/10.4028/www.scientific.net/amm.55-57.101.
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The Von Neumann Model on national economical system is investigated. A new discrete-time input-output model on national economic system based on the classic Von Neumann Model is provided and the stability of this kind of model is researched. This new system belongs to the singular system. By the new mathematic method, this singular linear system will not be converted into the general linear system. Finally, a sufficient stability condition under which the discrete-time singular Extended Von Neumann Model is admissible is proved.
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Haney, Matthew M. "Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case." GEOPHYSICS 72, no. 5 (September 2007): SM35—SM46. http://dx.doi.org/10.1190/1.2750639.
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Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model — that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable.
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Wu, Xiu Mei, Tao Zi Si, and Lei Jiang. "Stable Computer Control Algorithm of Von Neumann Model." Advanced Materials Research 634-638 (January 2013): 4026–29. http://dx.doi.org/10.4028/www.scientific.net/amr.634-638.4026.
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The problem of computer control algorithm for the singular Von Neumann input-output model is researched. A kind of new mathematic method is applied to study the singular systems without converting them into general systems. A kind of stability condition under which the singular input-output model is admissible is proved with the form of linear matrix inequality. Based on this, a new state feedback stability criterion is established. Then the formula of a desired state feedback controller is derived.
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WESSELING, P. "von Neumann stability conditions for the convection-diffusion eqation." IMA Journal of Numerical Analysis 16, no. 4 (1996): 583–98. http://dx.doi.org/10.1093/imanum/16.4.583.
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M. Jemimah, M. Alpha, and Abubakar Alkasim. "Exact Solution of Couple Burgers’ Equation using Cubic B-Spline Collocation Method for Fluid Suspension/Colloid under the Influence of Gravity." African Journal of Advances in Science and Technology Research 14, no. 1 (April 30, 2024): 73–85. http://dx.doi.org/10.62154/ymthy538.
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In this research, Cubic B-Spline plane method was used to solve numerically the one-dimensional Burger’s Equation with initial condition , boundary conditions; . The cubic trigonometric B-spline was used for interpolating the solutions at each time and using the Von-Neumann method to check the stability. The obtained numerical result showed that the method was efficient, robust and reliable for solving Burgers’ Equation accurately even involving high Reynolds numbers for which the exact solutions have failed.
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Hafiz, Khandaker Md Eusha Bin, and Laek Sazzad Andallah. "Second Order Scheme For Korteweg-De Vries (KDV) Equation." Journal of Bangladesh Academy of Sciences 43, no. 1 (July 16, 2019): 85–93. http://dx.doi.org/10.3329/jbas.v43i1.42237.
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The kinematics of the solitary waves is formed by Korteweg-de Vries (KdV) equation. In this paper, a third order general form of the KdV equation with convection and dispersion terms is considered. Explicit finite difference schemes for the numerical solution of the KdV equation is investigated and stability condition for a first-order scheme using convex combination method is determined. Von Neumann stability analysis is performed to determine the stability condition for a second order scheme. The well-known qualitative behavior of the KdV equation is verified and error estimation for comparisons is performed.
Journal of Bangladesh Academy of Sciences, Vol. 43, No. 1, 85-93, 2019
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Quintana-Murillo, J., and S. B. Yuste. "An Explicit Numerical Method for the Fractional Cable Equation." International Journal of Differential Equations 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/231920.
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An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations.
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Luiz, Kariston Stevan, Juniormar Organista, Eliandro Rodrigues Cirilo, Neyva Maria Lopes Romeiro, and Paulo Laerte Natti. "Numerical convergence of a Telegraph Predator-Prey system." Semina: Ciências Exatas e Tecnológicas 43, no. 1Esp (November 30, 2022): 51–66. http://dx.doi.org/10.5433/1679-0375.2022v43n1espp51.
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Numerical convergence of a Telegraph Predator-Prey system is studied. This partial differential equation (PDE) system can describe various biological systems with reactive, diffusive, and delay effects. Initially, the PDE system was discretized by the Finite Differences method. Then, a system of equations in a time-explicit form and in a space-implicit form was obtained. The consistency of the Telegraph Predator-Prey system discretization was verified. Von Neumann stability conditions were calculated for a Predator-Prey system with reactive terms and for a Delayed Telegraph system. On the other hand, for our Telegraph Predator-Prey system, it was not possible to obtain the von Neumann conditions analytically. In this context, numerical experiments were carried out and it was verified that the mesh refinement and the model parameters, reactive constants, diffusion coefficients and delay constants, determine the stability/instability conditions of the discretized equations. The results of numerical experiments were presented.
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Erkurşun Özcan, Nazife. "On ergodic properties of operator nets on the predual of von neumann algebras." Studia Scientiarum Mathematicarum Hungarica 55, no. 4 (December 2018): 479–86. http://dx.doi.org/10.1556/012.2018.55.4.1414.
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In this paper, we proved theorems which give the conditions that special operator nets on a predual of von Neumann algebras are strongly convergent under the Markov case. Moreover, we investigate asymptotic stability and existence of a lower-bound function for such nets.
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Fukuyo, Kazuhiro. "Conditional stability of Larkin methods with non-uniform grids." Theoretical and Applied Mechanics 37, no. 2 (2010): 139–59. http://dx.doi.org/10.2298/tam1002139f.
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Stability analysis based on the von Neumann method showed that the Larkin methods for two-dimensional heat conduction with non- uniform grids are conditionally stable while they are known to be unconditionally stable with uniform grids. The stability criteria consisting of the dimensionless time step ?t, the space intervals ?x, ?y, and the ratios of neighboring space intervals ?, ? were derived from the stability analysis. A subsequent numerical experiment demonstrated that solutions derived by the Larkin methods with non-uniform grids lose stability and accuracy when the criteria are not satisfied.
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Larkin, Eugene, Alexey Bogomolov, and Sergey Feofilov. "Stability of digital feedback control systems." MATEC Web of Conferences 161 (2018): 02004. http://dx.doi.org/10.1051/matecconf/201816102004.
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Specific problems arising, when Von Neumann type computer is used as feedback element, are considered. It is shown, that due to specifics of operation this element introduce pure lag into control loop, and lag time depends on complexity of algorithm of control. Method of evaluation of runtime between reading data from sensors of object under control and write out data to actuator based on the theory of semi- Markov process is proposed. Formulae for time characteristics estimation are obtained. Lag time characteristics are used for investigation of stability of linear systems. Digital PID controller is divided onto linear part, which is realized with a soft and pure lag unit, which is realized with both hardware and software. With use notions amplitude and phase margins, condition for stability of system functioning are obtained. Theoretical results are confirm with computer experiment carried out on the third-order system.
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Lezani, Nadine Mulya, and Ummu Habibah. "Numerical Solution of Burgers Equation using Cubic B-Spline Collocation Method and Neumann Boundary Condition." Indonesian Journal of Mathematics and Applications 1, no. 2 (September 30, 2023): 25–34. http://dx.doi.org/10.21776/ub.ijma.2023.001.02.3.
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Burgers equation is one of the nonlinear differential equations which is generally difficult to determine its analytical solution, so it is necessary to do a numerical approach. This article discusses the numerical solution of the Burgers equation using the Cubic B-Spline Collocation method. The first step is to derive the numerical scheme using the Cubic B-Spline Collocation method for the space variable and the Crank-Nicholson method for the time variable. Furthermore, based on von Neumann stability analysis, it is obtained that the numerical scheme of Burgers equation is unconditionally stable. By performing numerical simulations using different and step sizes, it can be shown that the absolute value of the resulting error will be smaller for the step sizes of and which is getting smaller.
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Krivovichev, Gerasim Vladimirovich. "On the stability of lattice boltzmann equations for one-dimensional diffusion equation." International Journal of Modeling, Simulation, and Scientific Computing 08, no. 01 (January 10, 2017): 1750013. http://dx.doi.org/10.1142/s1793962317500131.
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Stability analysis of lattice Boltzmann equations (LBEs) on initial conditions for one-dimensional diffusion is performed. Stability of the solution of the Cauchy problem for the system of linear Bhatnaghar–Gross–Krook kinetic equations is demonstrated for the cases of D1Q2 and D1Q3 lattices. Stability of the scheme for D1Q2 lattice is analytically analyzed by the method of differential approximation. Stability of parametrical scheme is numerically investigated by von Neumann method in parameter space. As a result of numerical analysis, the correction of the hypothesis on transfer of stability conditions of the scheme for macroequation to the system of LBEs is demonstrated.
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Castillo, Paul, and Sergio Gómez. "Análisis de Von Neumann para el métodoLocal Discontinuous Galerkin en 1D." Revista Integración 37, no. 2 (August 2, 2019): 199–217. http://dx.doi.org/10.18273/revint.v37n2-2019001.
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Using the von Neumann analysis as a theoretical tool, an analysisof the stability conditions of some explicit time marching schemes, in com-bination with the spatial discretizationLocal Discontinuous Galerkin(LDG)and high order approximations, is presented. The stabilityconstant, CFL(Courant-Friedrichs-Lewy), is studied as a function of theLDG parametersand the approximation degree. A series of numerical experiments is carriedout to validate the theoretical results.
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Owino, Benard, Fredrick Nyamwala, and David Ambogo. "Stability of Krein-von Neumann self-adjoint operator extension under unbounded perturbations." Annals of Mathematics and Computer Science 23 (April 26, 2024): 29–47. http://dx.doi.org/10.56947/amcs.v23.300.
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We have considered a fourth order difference operator defined on the Hilbert space of square summable sequences on N. We investigated the stability of existence of Krein-von Neumann self-adjoint extension of difference operators under bounded and unbounded coefficients. Using asymptotic summation based on discretised Levinson's theorem and appropriate smoothness and decay conditions, we have shown that unlike the case of deficiency indices and discrete spectrum, the existence of positive self-adjoint operator extensions is stable under unbounded perturbations. These results now exhaustively characterise the spectral and structural properties of difference operators with unbounded coefficients.
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Raed, Raed. "On The Numerical Solutions of the Neutrosophic One-Dimensional Sine-Gordon System." International Journal of Neutrosophic Science 25, no. 1 (2025): 25–36. http://dx.doi.org/10.54216/ijns.250303.
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This paper uses finite difference methods to study the numerical solution for neutrosophic Sine-Gordon system in one dimension. We use the explicit method and Crank-Nicholson method. Also, an effective comparison between the results of the two methods has been made, where we obtain the result that Crank-Nicholson method is more accurate than the explicit method, but the explicit method is easier. We also study the stability analysis for each method by using Fourier (Von-Neumann) method and get that Crank-Nicholson method is unconditionally stable while the Explicit method is stable under the condition 𝑟2≤1𝑐2 and 𝑟2≤1.
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Liu, Yingfan. "An Optimal Lower Eigenvalue System." Abstract and Applied Analysis 2011 (2011): 1–20. http://dx.doi.org/10.1155/2011/208624.
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An optimal lower eigenvalue system is studied, and main theorems including a series of necessary and suffcient conditions concerning existence and a Lipschitz continuity result concerning stability are obtained. As applications, solvability results to some von-Neumann-type input-output inequalities, growth, and optimal growth factors, as well as Leontief-type balanced and optimal balanced growth paths, are also gotten.
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Mousa, Mohamed M., and Wen-Xiu Ma. "Efficient modeling of shallow water equations using method of lines and artificial viscosity." Modern Physics Letters B 34, no. 04 (December 19, 2019): 2050051. http://dx.doi.org/10.1142/s0217984920500517.
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In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results.
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PAN, X. F., AIGUO XU, GUANGCAI ZHANG, and SONG JIANG. "LATTICE BOLTZMANN APPROACH TO HIGH-SPEED COMPRESSIBLE FLOWS." International Journal of Modern Physics C 18, no. 11 (November 2007): 1747–64. http://dx.doi.org/10.1142/s0129183107011716.
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We present an improved lattice Boltzmann model for high-speed compressible flows. The model is composed of a discrete-velocity model by Kataoka and Tsutahara15 and an appropriate finite-difference scheme combined with an additional dissipation term. With the dissipation term parameters in the model can be flexibly chosen so that the von Neumann stability condition is satisfied. The influence of the various model parameters on the numerical stability is analyzed and some reference values of parameter are suggested. The new scheme works for both subsonic and supersonic flows with a Mach number up to 30 (or higher), which is validated by well-known benchmark tests. Simulations on Riemann problems with very high ratios (1000:1) of pressure and density also show good accuracy and stability. Successful recovering of regular and double Mach shock reflections shows the potential application of the lattice Boltzmann model to fluid systems where non-equilibrium processes are intrinsic. The new scheme for stability can be easily extended to other lattice Boltzmann models.
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O’Brien, Gareth S. "3D rotated and standard staggered finite-difference solutions to Biot’s poroelastic wave equations: Stability condition and dispersion analysis." GEOPHYSICS 75, no. 4 (July 2010): T111—T119. http://dx.doi.org/10.1190/1.3432759.
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A fourth-order in space and second-order in time 3D staggered (SG) and rotated-staggered-grid (RSG) method for the solution of Biot’s equation are presented. The numerical dispersion and stability conditions are derived using a von Neumann analysis. The exact stability condition is calculated from the roots of a 12th-order polynomial and therefore no nontrivial expression exists. To overcome this, a 1D stability condition is usually generalized to three dimensions. It is shown that in certain cases, the 1D approximate stability condition is violated by a 3D SG method. The RSG method obeys the approximate 1D stability condition for the material properties and spatiotemporal scales in the examples shown. Both methods have been verified against an analytical solution for an infinite homogeneous porous medium with a misfit error of less than 0.5%. A free surface has been implement-ed to test the accuracy of this boundary condition. It also serves as a test of the methods to include high material contrasts. The methods have been compared with a quasi-analytical solution. For the specific material properties, spatial grid scaling, and propagation distance used in the test, a maximum error of 3.5% for the SG and 4.1% for the RSG was found. These errors depend on the propagation distance, temporal and spatial scales, and accuracy of the quasi-analytical solution. No discernable difference was found between the two methods except for time steps comparable with the stability-criteria threshold time step, the SG was found to be unstable. However, the RSG remained stable for a homogeneous half-space. Time steps, comparable to the stability criteria, reduce the computational time at the cost of a reduction in accuracy. The methods allow wave propagation to be modeled in a porous medium with a free surface.
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Agnes, Agnes. "On the Numerical Solutions Based On Exponential Finite Difference Method for Kuramoto-Sivashinsky Equation and Numerical Stability Analysis." Neutrosophic and Information Fusion 4, no. 2 (2024): 30–44. http://dx.doi.org/10.54216/nif.040204.
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In this paper, we solve the Kuramoto-Sivashinsky Equation numerically by finite-difference methods, using two different schemes which are the Fully Implicit scheme and Exponential finite difference scheme, because of the existence of the fourth derivative in the equation we suggested a treatment for the numerical solution of the two previous scheme by parting the mesh grid into five regions, the first region represents the first boundary condition, the second at the grid point x1, while the third represents the grid points x2,x3,…xn-2, the fourth represents the grid point xn-1 and the fifth is the second boundary condition. We also, study the numerical stability by Fourier (Von-Neumann) method for the two scheme which used in the solution on all mesh points to ensure the stability of the point which had been treated in the suggested style, we using two interval with two initial condition and the numerical results obtained by using these schemes are compare with Exact Solution of Equation Excellent approximate is found between the Exact Solution and numerical Solutions of these methods.
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Kim, Dojin. "A Modified PML Acoustic Wave Equation." Symmetry 11, no. 2 (February 2, 2019): 177. http://dx.doi.org/10.3390/sym11020177.
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In this paper, we consider a two-dimensional acoustic wave equation in an unbounded domain and introduce a modified model of the classical un-split perfectly matched layer (PML). We apply a regularization technique to a lower order regularity term employed in the auxiliary variable in the classical PML model. In addition, we propose a staggered finite difference method for discretizing the regularized system. The regularized system and numerical solution are analyzed in terms of the well-posedness and stability with the standard Galerkin method and von Neumann stability analysis, respectively. In particular, the existence and uniqueness of the solution for the regularized system are proved and the Courant-Friedrichs-Lewy (CFL) condition of the staggered finite difference method is determined. To support the theoretical results, we demonstrate a non-reflection property of acoustic waves in the layers.
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Lines, Larry R., Raphael Slawinski, and R. Phillip Bording. "A recipe for stability of finite‐difference wave‐equation computations." GEOPHYSICS 64, no. 3 (May 1999): 967–69. http://dx.doi.org/10.1190/1.1444605.
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Finite‐difference solutions to the wave equation are pervasive in the modeling of seismic wave propagation (Kelly and Marfurt, 1990) and in seismic imaging (Bording and Lines, 1997). That is, they are useful for the forward problem (modeling) and the inverse problem (migration). In computational solutions to the wave equation, it is necessary to be aware of conditions for numerical stability. In this short note, we examine a convenient recipe for insuring stability in our finite‐difference solutions to the wave equation. The stability analysis for finite‐difference solutions of partial differential equations is handled using a method originally developed by Von Neumann and described by Press et al. (1986, p. 827–830).
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Todor, Radu, Ionuţ Chiose, and George Marinescu. "Morse inequalities for covering manifolds." Nagoya Mathematical Journal 163 (September 2001): 145–65. http://dx.doi.org/10.1017/s0027763000007947.
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We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.
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Hutchinson, A. J., C. Harley, and E. Momoniat. "Numerical Investigation of the Steady State of a Driven Thin Film Equation." Journal of Applied Mathematics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/181939.
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A third-order ordinary differential equation with application in the flow of a thin liquid film is considered. The boundary conditions come from Tanner's problem for the surface tension driven flow of a thin film. Symmetric and nonsymmetric finite difference schemes are implemented in order to obtain steady state solutions. We show that a central difference approximation to the third derivative in the model equation produces a solution curve with oscillations. A difference scheme based on a combination of forward and backward differences produces a smooth accurate solution curve. The stability of these schemes is analysed through the use of a von Neumann stability analysis.
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BHATTACHARYA, ANINDYA, and AMIT K. BISWAS. "STABILITY OF THE CORE IN A CLASS OF NTU GAMES: A CHARACTERIZATION." International Game Theory Review 04, no. 02 (June 2002): 165–72. http://dx.doi.org/10.1142/s0219198902000628.
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The core and the stable set are possibly the two most crucially important solution concepts for cooperative games. The relation between the two has been investigated in the context of symmetric transferable utility games and this has been related to the notion of large core. In this paper the relation between the von-Neumann–Morgenstern stability of the core and the largeness of it is investigated in the case of non-transferable utility (NTU) games. The main findings are that under certain regularity conditions, if the core of an NTU game is large then it is a stable set and for symmetric NTU games the core is a stable set if and only if it is large.
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Mugheri, Abdul Qadir. "Numerical Simulation of One-Dimensional Advection Diffusion Equation by New Hybrid Explicit Finite Difference Schemes." Volume 21, Issue 1 21, no. 1 (June 30, 2023): 54–62. http://dx.doi.org/10.52584/qrj.2101.07.
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In this paper, the aim is to develop two Hybrid Explicit Schemes based on the Finite Difference method for a one- dimensional Advection Diffusion Equation. Moreover, the study considered the advection-diffusion equation as an initial boundary value problem (IBVP) for numerical solutions obtained from various second-order explicit methods along with the solution by proposed methods. Von-Neumann stability analysis is used to analyze the stability of the developed schemes graphically. In the numerical analysis of errors, the L 2 has been computed to compare proposed methods with existing methods in literature and has been carried out in different conditions and step sizes. Proposed methods are robust explicit methods for the purpose of solution of the 1-D advection-diffusion equation.
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Nawaz, Yasir, Muhammad Shoaib Arif, Wasfi Shatanawi, and Muhammad Usman Ashraf. "A Fourth Order Numerical Scheme for Unsteady Mixed Convection Boundary Layer Flow: A Comparative Computational Study." Energies 15, no. 3 (January 27, 2022): 910. http://dx.doi.org/10.3390/en15030910.
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In this paper, a three-stage fourth-order numerical scheme is proposed. The first and second stages of the proposed scheme are explicit, whereas the third stage is implicit. A fourth-order compact scheme is considered to discretize space-involved terms. The stability of the fourth-order scheme in space and time is checked using the von Neumann stability criterion for the scalar case. The stability region obtained by the scheme is more than the one given by explicit Runge–Kutta methods. The convergence conditions are found for the system of partial differential equations, which are non-dimensional equations of heat transfer of Stokes first and second problems. The comparison of the proposed scheme is made with the existing Crank–Nicolson scheme. From this comparison, it can be concluded that the proposed scheme converges faster than the Crank–Nicolson scheme. It also produces less relative error than the Crank–Nicolson method for time-dependent problems.
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GROGGER, HERWIG A. "OPTIMIZED ARTIFICIAL DISSIPATION TERMS FOR ENHANCED STABILITY LIMITS." Journal of Computational Acoustics 15, no. 02 (June 2007): 235–53. http://dx.doi.org/10.1142/s0218396x07003329.
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Finite difference approximations for the convection equation are developed, which exhibit enhanced stability limits for explicit Runge–Kutta integration. Stability limits are increased by adding artificial dissipation terms, which are optimized to yield greatest stable time steps. For the artificial dissipation terms, symmetric finite difference approximations of even-order derivatives are used with differencing stencils equal to the convective stencils. The spatial discretization inclusive of the added dissipation term is shown to be consistent with a first derivative. The formal order of accuracy in space is decreased by one order, while the order of time integration is not affected. As a result, the time step limits of originally stable Runge–Kutta integration is increased, for some combinations of spatial discretization and time integration by a factor of two. Algorithms, which are unstable without damping are stabilized. The dispersion properties of the algorithms are not influenced by the proposed damping terms. Spectral analysis of the algorithms show very low dissipation error for dimensionless wave numbers k Δ x < 0.5. Stability conditions based on von Neumann stability analysis are given for the proposed schemes for explicit Runge–Kutta time integration of orders up to ten.
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Momoniat, E., M. M. Rashidi, and R. S. Herbst. "Numerical Investigation of Thin Film Spreading Driven by Surfactant Using Upwind Schemes." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/325132.
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Numerical solutions of a coupled system of nonlinear partial differential equations modelling the effects of surfactant on the spreading of a thin film on a horizontal substrate are investigated. A CFL condition is obtained from a von Neumann stability analysis of a linearised system of equations. Numerical solutions obtained from a Roe upwind scheme with a third-order TVD Runge-Kutta approximation to the time derivative are compared to solutions obtained with a Roe-Sweby scheme coupled to a minmod limiter and a TVD approximation to the time derivative. Results from both of these schemes are compared to a Roe upwind scheme and a BDF approximation to the time derivative. In all three cases high-order approximations to the spatial derivatives are employed on the interior points of the spatial domain. The Roe-BDF scheme is shown to be an efficient numerical scheme for capturing sharp changes in gradient in the free surface profile and surfactant concentration. Numerical simulations of an initial exponential free surface profile coupled with initial surfactant concentrations for both exogenous and endogenous surfactants are considered.
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Redouane, Kelthoum Lina, Nouria Arar, Abdellatif Ben Makhlouf, and Abeer Alhashash. "A Higher-Order Improved Runge–Kutta Method and Cubic B-Spline Approximation for the One-Dimensional Nonlinear RLW Equation." Mathematical Problems in Engineering 2023 (April 19, 2023): 1–13. http://dx.doi.org/10.1155/2023/4753873.
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This article developed a significant improvement of a Galerkin-type approximation to the regularized long-wave equation (RLW) solution under homogeneous Dirichlet boundary conditions for achieving higher accuracy in time variables. First, a basis derived from cubic B-splines and limit conditions is used to perform a Galerkin-type approximation. Then, a Crank–Nicolson and fourth-order 4-stage improved Runge–Kutta scheme (IRK4) is used to discretize time. Both a strong stability analysis of a fully discrete IRK4 scheme and the evaluation of Von Neumann stability of the proposed Crank–Nicolson technique are examined. We demonstrate the efficiency of our method with two test problems. The analytical and numerical solutions found in the literature are then contrasted with the approximate solutions produced by the suggested method. The validated numerical results illustrate that the provided technique is more efficient and converges faster than earlier research, resulting in less computational time, smaller space dimensions, and storage. As a result, the proposed numerical approach is appealing for approximating PDEs whose explicit solution is unknown for a variety of boundary conditions.
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Al-Khateeb, Areen. "Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods." Axioms 13, no. 4 (March 26, 2024): 221. http://dx.doi.org/10.3390/axioms13040221.
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In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of 0≤ ξ ≤ 1. The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example.
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Yokuş, Asıf. "Truncation and convergence dynamics: KdV Burgers model in the sense of Caputo derivative." Boletim da Sociedade Paranaense de Matemática 40 (January 26, 2022): 1–7. http://dx.doi.org/10.5269/bspm.47472.
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This study examines the time fractional KdV Burgers equation with the initial conditions by using the extended result on Caputo formula, finite difference method (FDM). For this reason, various fractional differential operators are defined and analyzed. In order to check the stability of the numerical scheme, the Fourier-von Neumann technique is used. By presenting an example of KdV Burgers equation above mentioned issues are discussed and numerical solutions of the error estimates have been found for the FDM. For the errors in $L_2$ and $L_\infty$ the method accuracy has been controlled. Moreover, the obtained results have been compared with the exact solution for different cases of non-integer order and the behavior of the potentials u is presented as a graph. The numerical results have been shown in tables.
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Baber, Muhammad Zafarullah, Nauman Ahmed, Muhammad Waqas Yasin, Muhammad Sajid Iqbal, Ali Akgül, Alicia Cordero, and Juan R. Torregrosa. "Comparisons of Numerical and Solitary Wave Solutions for the Stochastic Reaction–Diffusion Biofilm Model including Quorum Sensing." Mathematics 12, no. 9 (April 24, 2024): 1293. http://dx.doi.org/10.3390/math12091293.
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This study deals with a stochastic reaction–diffusion biofilm model under quorum sensing. Quorum sensing is a process of communication between cells that permits bacterial communication about cell density and alterations in gene expression. This model produces two results: the bacterial concentration, which over time demonstrates the development and decomposition of the biofilm, and the biofilm bacteria collaboration, which demonstrates the potency of resistance and defense against environmental stimuli. In this study, we investigate numerical solutions and exact solitary wave solutions with the presence of randomness. The finite difference scheme is proposed for the sake of numerical solutions while the generalized Riccati equation mapping method is applied to construct exact solitary wave solutions. The numerical scheme is analyzed by checking consistency and stability. The consistency of the scheme is gained under the mean square sense while the stability condition is gained by the help of the Von Neumann criteria. Exact stochastic solitary wave solutions are constructed in the form of hyperbolic, trigonometric, and rational forms. Some solutions are plots in 3D and 2D form to show dark, bright and solitary wave solutions and the effects of noise as well. Mainly, the numerical results are compared with the exact solitary wave solutions with the help of unique physical problems. The comparison plots are dispatched in three dimensions and line representations as well as by selecting different values of parameters.
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Aliyi, Kedir, and Hailu Muleta. "Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation." Indian Journal of Advanced Mathematics 1, no. 2 (October 10, 2021): 4–14. http://dx.doi.org/10.35940/ijam.b1103.101221.
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In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘ ’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, N t , E and rate of convergence, N t , P . The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
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Aliyi, Kedir, and Hailu Muleta. "Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation." Indian Journal of Advanced Mathematics 1, no. 2 (October 10, 2021): 4–14. http://dx.doi.org/10.54105/ijam.b1103.101221.
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In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘ε’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, EN,Δt and rate of convergence, Pε N,Δt. The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
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Fournié, Michel, and Alain Rigal. "High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows." Communications in Computational Physics 9, no. 4 (April 2011): 994–1019. http://dx.doi.org/10.4208/cicp.230709.080710a.
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AbstractWithin the projection schemes for the incompressible Navier-Stokes equations (namely “pressure-correction” method), we consider the simplest method (of order one in time) which takes into account the pressure in both steps of the splitting scheme. For this scheme, we construct, analyze and implement a new high order compact spatial approximation on nonstaggered grids. This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions (without error on the velocity) which could be extended to more general splitting. We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis. Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions. Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations (including the driven cavity benchmark) to illustrate the theoretical results.
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Sochacki, James, Robert Kubichek, John George, W. R. Fletcher, and Scott Smithson. "Absorbing boundary conditions and surface waves." GEOPHYSICS 52, no. 1 (January 1987): 60–71. http://dx.doi.org/10.1190/1.1442241.
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One of the major problems in numerically simulating waves traveling in the Earth is that an artificial boundary must be introduced to produce unique solutions. To eliminate the spurious reflections introduced by this artificial boundary, we use a damping expression based on analogies to shock absorbers. This method can reduce the amplitude of the reflected wave to any pre‐specified value and is successful for waves at any angle of incidence. The method can eliminate unwanted reflections from the surface, reflections at the corners of the model, and waves reflected off an interface that strike the artificial boundary. Many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary. Stability problems often occur with these boundary conditions. The method we use at the artificial boundary allows use of stable Dirichlet or von Neumann conditions. Since the surface motion is easily measured when the waves are induced by normal seismic techniques, approximations of surface waves are needed to obtain information on Rayleigh waves. An implicit finite‐difference scheme that is relatively easy to incorporate into existing numerical simulators is used to obtain the surface data for the forward finite‐difference approximations.
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Ginzburg, Irina. "Truncation Errors, Exact And Heuristic Stability Analysis Of Two-Relaxation-Times Lattice Boltzmann Schemes For Anisotropic Advection-Diffusion Equation." Communications in Computational Physics 11, no. 5 (May 2012): 1439–502. http://dx.doi.org/10.4208/cicp.211210.280611a.
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AbstractThis paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the two-relaxation-times (TRT) collision operator. First we propose a simple method to derive the truncation errors from the exact, central-difference type, recurrence equations of the TRT scheme. They also supply its equivalent three-time-level discretization form. Two different relationships of the two relaxation rates nullify the third (advection) and fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order advection-d if fusion error, because of the intrinsic fourth-order numerical diffusion. The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils. The sufficient stability conditions are proposed for the most stable (OTRT) family, which enables modeling at any Peclet numbers with the same velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates, in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill.
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Huntul, M. J. "Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions." Physica Scripta 97, no. 3 (February 23, 2022): 035004. http://dx.doi.org/10.1088/1402-4896/ac54d0.
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Abstract In this paper, we considered an inverse problem of recovering the space-dependent source coefficient in the third-order pseudo-parabolic equation from final over-determination condition. This inverse problem appears extensively in the modelling of various phenomena in physics such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, etc. The unique solvability theorem for this inverse problem is supplied. However, since the governing equation is yet ill-posed (very slight errors in the final input may cause relatively significant errors in the output source term), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown force term. The third-order pseudo-parabolic problem is discretized using the Cubic B-spline (CB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Both perturbed data and analytical solutions are inverted. Numerical outcomes are reported and discussed. The computational efficiency of the method is investigated by small values of CPU time. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.
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Devi, Rekha, and Shilpa Sood. "Numerical Investigation of Three-Dimensional Magnetohydrodynamic Flow of Ag H2O Nanofluid Over an Oscillating Surface in a Rotating Porous Medium." Indian Journal Of Science And Technology 17, no. 8 (February 15, 2024): 679–90. http://dx.doi.org/10.17485/ijst/v17i8.2892.
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Objective: To investigate the three-dimensional flow of a nanofluid (Ag-water) over a stretchable vertical oscillatory sheet. This study involves considering fluctuating temperatures on the sheet and comparing them to the free stream temperature. The formulation of the unsteady boundary layer equations leading to the flow of nanofluid also takes into consideration the occurrence of the heterogeneous-homogeneous chemical reaction and thermal radiation. Method: The governing equations and the boundary conditions have been derived in a dimensionless form by using the appropriate transformations, and they are then solved using an EFDS (Explicit Finite Difference Scheme) in Matlab software. The Von-Neumann stability analysis is used to determine the method’s stability requirements for constant sizes of the grid. Findings: The physical factors impact on the concentration fields, temperature distribution, and velocity distribution were obtained and are studied by graphs and described in extensive detail. Convergence and stability requirements are attained in order to achieve accurate solutions. Novelty: In this study fluctuations in the temperature and stretching velocity of sheet on three-dimensional magnetohydrodynamic flow of Ag − H2O nanofluid over an oscillating surface through rotating porous are taken into account. Impacts of porous media permeability, velocity slip, magnetic fields, nanoparticle volume fraction, heat radiation, rotation, and homogeneous and heterogeneous chemical reaction parameters had all been attempted to be determined. Keywords: Oscillatory Surface, Heat transmission, Nonlinear PDE, Explicit Finite Difference Scheme, Nanoparticle
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SHORT, M., I. I. ANGUELOVA, T. D. ASLAM, J. B. BDZIL, A. K. HENRICK, and G. J. SHARPE. "Stability of detonations for an idealized condensed-phase model." Journal of Fluid Mechanics 595 (January 8, 2008): 45–82. http://dx.doi.org/10.1017/s0022112007008750.
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The stability of travelling wave Chapman–Jouguet and moderately overdriven detonations of Zeldovich–von Neumann–Döring type is formulated for a general system that incorporates the idealized gas and condensed-phase (liquid or solid) detonation models. The general model consists of a two-component mixture with a one-step irreversible reaction between reactant and product. The reaction rate has both temperature and pressure sensitivities and has a variable reaction order. The idealized condensed-phase model assumes a pressure-sensitive reaction rate, a constant-γ caloric equation of state for an ideal fluid, with the isentropic derivative γ=3, and invokes the strong shock limit. A linear stability analysis of the steady, planar, ZND detonation wave for the general model is conducted using a normal-mode approach. An asymptotic analysis of the eigenmode structure at the end of the reaction zone is conducted, and spatial boundedness (closure) conditions formally derived, whose precise form depends on the magnitude of the detonation overdrive and reaction order. A scaling analysis of the transonic flow region for Chapman–Jouguet detonations is also studied to illustrate the validity of the linearization for Chapman–Jouguet detonations. Neutral stability boundaries are calculated for the idealized condensed-phase model for one- and two-dimensional perturbations. Comparisons of the growth rates and frequencies predicted by the normal-mode analysis for an unstable detonation are made with a numerical solution of the reactive Euler equations. The numerical calculations are conducted using a new, high-order algorithm that employs a shock-fitting strategy, an approach that has significant advantages over standard shock-capturing methods for calculating unstable detonations. For the idealized condensed-phase model, nonlinear numerical solutions are also obtained to study the long-time behaviour of one- and two-dimensional unstable Chapman–Jouguet ZND waves.
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Yokus, Asif, Bülent Kuzu, and Uğur Demiroğlu. "Investigation of solitary wave solutions for the (3 + 1)-dimensional Zakharov–Kuznetsov equation." International Journal of Modern Physics B 33, no. 29 (November 20, 2019): 1950350. http://dx.doi.org/10.1142/s0217979219503508.
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In this paper, the new traveling wave solutions containing the trigonometric functions, hyperbolic functions and rational functions of [Formula: see text]-dimensional Zakharov–Kuznetsov equation are obtained. The graphs of the solution functions are presented by giving specific values to the constants. Numerical solutions are obtained by using finite difference method with new initial condition. Von Neumann’s Stability, Consistency and Linear Stability analysis of the equation are performed and [Formula: see text], [Formula: see text] norm errors are also examined with the truncation error. The exact solution obtained is presented via numerical solutions and absolute error graphs, and the analysis of exact solution and the numerical solutions are performed. Complex operations and graphical drawings were made using the computer package program.
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Кривовичев, Г. В., and М. П. Мащинская. "Stability analysis of the implicit finite-difference-based upwind lattice Boltzmann schemes." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 2 (March 28, 2019): 116–27. http://dx.doi.org/10.26089/nummet.v20r212.
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Статья посвящена анализу устойчивости неявных конечно-разностных схем для системы кинетических уравнений, применяемых для проведения гидродинамических расчетов в рамках метода решеточных уравнений Больцмана. Представлены семейства двухслойных и трехслойных схем с направленными разностями первого-четвертого порядков аппроксимации по пространственным переменным. Важной особенностью схем является то, что конвективные слагаемые аппроксимируются одной конечной разностью. Показано, что в выражении для аппроксимационной вязкости схем высоких порядков отсутствуют фиктивные слагаемые, что позволяет применять их во всем диапазоне значений времени релаксации. Анализ устойчивости проводится по линейному приближению с использованием метода Неймана. Получены приближенные условия устойчивости в виде неравенств на значения параметра Куранта. При расчетах показано, что площади областей устойчивости в пространстве параметров у двухслойных схем больше, чем у трехслойных. Исследованные схемы могут применяться при расчетах как непосредственно, так и в методах типа предиктор-корректор. The paper is devoted to the stability analysis of the implicit finite-difference schemes for the system of kinetic equations used for the hydrodynamic computations in the framework of the lattice Boltzmann method. The families of two- and three-layer upwind schemes of the first to fourth approximation orders on spatial variables are considered. An important feature of the presented schemes is that the convective terms are approximated by one finite difference. It is shown that, for the high-order schemes, in the expression for the current viscosity there are no fictitious terms, which makes it possible to perform computations in the whole range of relaxation time values. The stability analysis is based on the application of the von Neumann method to the linear approximations of the schemes. The stability conditions are obtained in the form of inequalities imposed on the Courant number values. It is also shown that the areas of stability domains for the two-layer schemes are greater than for the three-layer schemes in the parameter space. The considered schemes can be used as the fully implicit schemes in computational algorithms directly or in the predictor-corrector methods.
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Zureigat, Hamzeh, Saleh Alshammari, Mohammad Alshammari, Mohammed Al-Smadi, and M. Mossa Al-Sawallah. "An in-depth examination of the fuzzy fractional cancer tumor model and its numerical solution by implicit finite difference method." PLOS ONE 19, no. 12 (December 20, 2024): e0303891. https://doi.org/10.1371/journal.pone.0303891.
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The cancer tumor model serves a s a crucial instrument for understanding the behavior of different cancer tumors. Researchers have employed fractional differential equations to describe these models. In the context of time fractional cancer tumor models, there’s a need to introduce fuzzy quantities instead of crisp quantities to accommodate the inherent uncertainty and imprecision in this model, giving rise to a formulation known as fuzzy time fractional cancer tumor models. In this study, we have developed an implicit finite difference method to solve a fuzzy time-fractional cancer tumor model. Instead of utilizing classical time derivatives in fuzzy cancer models, we have examined the effect of employing fuzzy time-fractional derivatives. To assess the stability of our proposed model, we applied the von Neumann method, considering the cancer cell killing rate as time-dependent and utilizing Caputo’s derivative for the time-fractional derivative. Additionally, we conducted various numerical experiments to assess the viability of this new approach and explore relevant aspects. Furthermore, our study identified specific needs in researching the cancer tumor model with fuzzy fractional derivative, aiming to enhance our inclusive understanding of tumor behavior by considering diverse fuzzy cases for the model’s initial conditions. It was found that the presented approach provides the ability to encompass all scenarios for the fuzzy time fractional cancer tumor model and handle all potential cases specifically focusing on scenarios where the net cell-killing rate is time-dependent.
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Lee, Hyoung In, and El Hang Lee. "The Minimum Wave Damping Selects the Most Favored Solution from Multiple Ones to Acoustic-Like Problems." Materials Science Forum 673 (January 2011): 11–20. http://dx.doi.org/10.4028/www.scientific.net/msf.673.11.
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Back in 1990, D. S. Stewart and the first author contributed significantly to understanding the one-dimensional stability of detonation waves [1]. For this purpose, the reactive Euler’s equation with the one-component reaction term was linearized around the steady state of the well-known ZND (Zeldovich-Doering-von Neumann) model. The key aspect of this paper was to derive the linearized radiation condition (named after A. Sommerfeld). They numerically found multiple eigenvalues for pairs of the temporal frequency and temporal attenuation rate (TAR). Of course, the propagating-wave mode having the least value of the TAR (in the sense of its absolute value) was selected. The successful numerical implementation of the far-field radiation condition is a must when it comes to incorporating a large surrounding space into a problem of finite extent. To one of the sure examples in this category belong the problems involving detonation waves, where high-energy-rate processes take place in spatially confined spaces while the surrounding space should be taken into account for reasons of energy loss (or leaky waves in the language of optics). In another fascinating area of science is nano-photonics, where energy transport should be handled in highly confined regions of space, yet being surrounded by unbounded (dielectric) media. The total energy release in nano-photonics is certainly much smaller than that involved in detonation. However, the energy per unit nanometer-scale volume is not negligibly small in nano-photonics. Over the years, the first author has been successful in implementing both theory and numerical methods to find a multitude of eigenvalues in optics [2]. In this case, the governing Maxwell’s equations are already in a linearized form, being in a sense similar to the linearized Euler equations. In addition, the noble metals such as gold and silver are instrumental in enhancing local electric-field intensities, for which the science of plasmonics is being vigorously investigated in nano-photonics. Even the Bloch’s hydrodynamic equation describing the collective motion of the electrons is akin to the Navier-Stokes equations [3]. Meanwhile, assuming a real-valued frequency has been an old tradition in optics, partly because the real-valued photon’s energy is proportional to frequency and normally the continuous-wave (cw) approximation holds true. In a radical departure from this optical scientists’ tradition, we have recently attempted to deal with complex-valued frequencies in examining the wave propagations around nanoparticles [4, 5]. In consequence, the stability of multiple propagating waves was successfully determined for selecting most realizable wave mode. Further interesting points of the interplay between the two seemingly disparate branches of science (fluid dynamics and photonics) will be expounded in this talk.
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Peris, Josep E., and Begoña Subiza. "A reformulation of von Neumann–Morgenstern stability: -stability." Mathematical Social Sciences 66, no. 1 (July 2013): 51–55. http://dx.doi.org/10.1016/j.mathsocsci.2013.01.001.
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Kawasaki, Ryo. "Roth–Postlewaite stability and von Neumann–Morgenstern stability." Journal of Mathematical Economics 58 (May 2015): 1–6. http://dx.doi.org/10.1016/j.jmateco.2015.02.002.
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Xu, Fang Qin, and Lei Jiang. "Modeling of Stochastic Von Neumann Model of Mobile Service." Key Engineering Materials 474-476 (April 2011): 11–14. http://dx.doi.org/10.4028/www.scientific.net/kem.474-476.11.
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A stochastic Von Neumann model to describe companies’ input-output process on mobile services is provided. Through the theories from input-output economics, an extended singular stochastic Von Neumann model on mobile service is researched. The problem of stability of this kind of stochastic Von Neumann model on mobile services is researched. A new mathematic method is applied to study the singular systems without converting them into general systems. The parameter uncertainties are considered. A new stability criterion for the extended stochastic Von Neumann model is given to ensure the stability of input-output model.