Journal articles: 'Type-depedent finite difference scheme'

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Cheng, Xiaohan, Jianhu Feng, Supei Zheng, and Xueli Song. "A new type of finite difference WENO schemes for Hamilton–Jacobi equations." International Journal of Modern Physics C 30, no. 02n03 (February 2019): 1950020. http://dx.doi.org/10.1142/s0129183119500207.

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In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.

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Vulanović, Relja. "An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems." Computational Methods in Applied Mathematics 4, no. 3 (2004): 368–83. http://dx.doi.org/10.2478/cmam-2004-0020.

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AbstractThe discretization meshes of the Shishkin type are more suitable for high- order finite-difference schemes than Bakhvalov-type meshes. This point is illustrated by the construction of a hybrid scheme for a class of semilinear singularly perturbed reaction-diffusion problems. A sixth-order five-point equidistant scheme is used at most of the mesh points inside the boundary layers, whereas lower-order three-point schemes are used elsewhere. It is proved under certain conditions that this combined scheme is almost sixth-order accurate and that its error does not increase when the perturbation parameter tends to zero.

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Hureuski, A. N. "Using IIR filters to build high-order finite difference schemes for the unsteady Schrödinger equation." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 4 (January 7, 2020): 413–24. http://dx.doi.org/10.29235/1561-2430-2019-55-4-413-424.

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High-order finite difference schemes for the time-dependent Schrödinger equation are investigated. Digital signal processing methods allowed proving the conservativeness of high-order finite difference schemes for the unsteady Schrödinger equation. The eighth-order scheme coefficients were found with the help of the proved theoretical results. The conditions for equivalence between the eighth-order finite difference scheme and the scheme in the form of a cascade of allpass first-order filters were found. The numerical analysis of the proposed scheme was made. It was shown that the high-order finite difference schemes gave better results on solving the linear Schrödinger equations comparing to the well-known fourthorder scheme on the six-point stencil, however, the high-order schemes in couple with the second-order splitting algorithm to the nonlinear Schrödinger equation do not lead to a radical improvement in the quality of numerical results. Practical issues implementing the proposed numerical technique are considered. The obtained results can be used to construct efficient solvers for linear and nonlinear Schrödinger-type equations by applying the splitting schemes of adequate accuracy order.

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Kettunen, Lauri, Jonni Lohi, Jukka Räbinä, Sanna Mönkölä, and Tuomo Rossi. "Generalized finite difference schemes with higher order Whitney forms." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 4 (July 2021): 1439–60. http://dx.doi.org/10.1051/m2an/2021026.

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Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee’s original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee’s scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee’s scheme for all problems of a class characterised on a Minkowski manifold by (i) a pair of first order partial differential equations and by (ii) a constitutive relation that couple the differential equations with a Hodge relation. In addition, we introduce a strategy to systematically exploit higher order Whitney elements in Yee-like approaches. This makes higher order interpolation possible both in time and space. For this, we show that Yee-like schemes preserve the local character of the Hodge relation, which is to say, the constitutive laws become imposed on a finite set of points instead of on all ordinary points of space. As a result, the usage of higher order Whitney forms does not compel to change the actual solution process at all. This is demonstrated with a simple example.

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Kim, Sung-Hoon, and Youn-sik Park. "An Improved Finite Difference Type Numerical Method for Structural Dynamic Analysis." Shock and Vibration 1, no. 6 (1994): 569–83. http://dx.doi.org/10.1155/1994/139352.

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An improved finite difference type numerical method to solve partial differential equations for one-dimensional (1-D) structure is proposed. This numerical scheme is a kind of a single-step, second-order accurate and implicit method. The stability, consistency, and convergence are examined analytically with a second-order hyperbolic partial differential equation. Since the proposed numerical scheme automatically satisfies the natural boundary conditions and at the same time, all the partial differential terms at boundary points are directly interpretable to their physical meanings, the proposed numerical scheme has merits in computing 1-D structural dynamic motion over the existing finite difference numeric methods. Using a numerical example, the suggested method was proven to be more accurate and effective than the well-known central difference method. The only limitation of this method is that it is applicable to only 1-D structure.

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Mohsen, A., H. El-Zoheiry, and L. Iskandar. "A highly accurate finite-difference scheme for a boussinesq-type equation." Applied Mathematics and Computation 55, no. 2-3 (May 1993): 201–12. http://dx.doi.org/10.1016/0096-3003(93)90021-6.

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Lee, Chun-Te, Jeng-Eng Lin, Chun-Che Lee, and Mei-Li Liu. "Some Remarks on the Stability Condition of Numerical Scheme of the KdV-type Equation." Journal of Mathematics Research 9, no. 4 (June 29, 2017): 11. http://dx.doi.org/10.5539/jmr.v9n4p11.

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This paper has employed a comparative study between the numerical scheme and stability condition. Numerical calculations are carried out based on three different numerical schemes, namely the central finite difference, fourier leap-frog, and fourier spectral RK4 schemes. Stability criteria for different numerical schemes are developed for the KdV equation, and numerical examples are put to test to illustrate the accuracy and stability between the solution profile and numerical scheme.

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Molavi-Arabshahi, Mahboubeh, and Zahra Saeidi. "Application of Compact Finite Difference Method for Solving Some Type of Fractional Derivative Equations." International Journal of Circuits, Systems and Signal Processing 15 (September 6, 2021): 1324–35. http://dx.doi.org/10.46300/9106.2021.15.143.

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In this paper, the compact finite difference scheme as unconditionally stable method is applied to some type of fractional derivative equation. We intend to solve with this scheme two kinds of a fractional derivative, first a fractional order system of Granwald-Letnikov type 1 for influenza and second fractional reaction sub diffusion equation. Also, we analyzed the stability of equilibrium points of this system. The convergence of the compact finite difference scheme in norm 2 are proved. Finally, various cases are used to test the numerical method. In comparison to other existing numerical methods, our results show that the scheme yields an accurate solution that is quick to compute.

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Koroglu, Canan, and Ayhan Aydin. "An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation." Advances in Mathematical Physics 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/4796070.

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A numerical solution of the modified Korteweg-de Vries (MKdV) equation is presented by using a nonstandard finite difference (NSFD) scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization. Local truncation error of the NSFD scheme and linear stability analysis are discussed. To test the accuracy and efficiency of the method, some numerical examples are given. The numerical results of NSFD scheme are compared with the exact solution and a standard finite difference scheme. The numerical results illustrate that the NSFD scheme is a robust numerical tool for the numerical integration of the MKdV equation.

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Dong, Haoyu, Changna Lu, and Hongwei Yang. "The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations." Mathematics 6, no. 10 (October 18, 2018): 211. http://dx.doi.org/10.3390/math6100211.

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We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

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Li, Qin, Qilong Guo, and Hanxin Zhang. "Analyses of the Dispersion Overshoot and Inverse Dissipation of the High-Order Finite Difference Scheme." Advances in Applied Mathematics and Mechanics 5, no. 06 (December 2013): 809–24. http://dx.doi.org/10.4208/aamm.2012.m5.

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AbstractAnalyses were performed on the dispersion overshoot and inverse dissipation of the high-order finite difference scheme using Fourier and precision analysis. Schemes under discussion included the pointwise- and staggered-grid type, and were presented in weighted form using candidate schemes with third-order accuracy and three-point stencil. All of these were commonly used in the construction of difference schemes. Criteria for the dispersion overshoot were presented and their critical states were discussed. Two kinds of instabilities were studied due to inverse dissipation, especially those that occur at lower wave numbers. Criteria for the occurrence were presented and the relationship of the two instabilities was discussed. Comparisons were made between the analytical results and the dispersion/dissipation relations by Fourier transformation of typical schemes. As an example, an application of the criteria was given for the remedy of inverse dissipation in Weirs & Martín’s third-order scheme.

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Pavelchuk, Anna Vladimirovna, and Anna Gennadievna Maslovskaya. "MODIFIED FINITE-DIFFERENCE SCHEME FOR SOLVING ONE CLASS OF CONVECTION-REACTION-DIFFUSION PROBLEMS." Messenger AmSU, no. 93 (2021): 7–14. http://dx.doi.org/10.22250/jasu.93.2.

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The paper reviews approaches to the construction of finite-difference methods for solving time-dependent diffusion equations and transport equations. A modified computational scheme for solving a semilinear multidimensional equation of the «reaction – diffusion – convection» type is presented. The hybrid computational scheme is based on the alternating directions method and the Robert-Weiss scheme.

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Zlotnik, A. A., and B. N. Chetverushkin. "Spectral Stability Conditions for an Explicit Three-Level Finite-Difference Scheme for a Multidimensional Transport Equation with Perturbations." Differential Equations 57, no. 7 (July 2021): 891–900. http://dx.doi.org/10.1134/s0012266121070065.

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Abstract We study difference schemes associated with a simplified linearized multidimensional hyperbolic quasi-gasdynamic system of differential equations. It is shown that an explicit two-level vector difference scheme with flux relaxation for a second-order hyperbolic equation with variable coefficients that is a perturbation of the transport equation with a parameter multiplying the highest derivatives can be reduced to an explicit three-level difference scheme. In the case of constant coefficients, the spectral condition for the time-uniform stability of this explicit three-level difference scheme is analyzed, and both sufficient and necessary conditions for this condition to hold are derived, in particular, in the form of Courant type conditions on the ratio of temporal and spatial steps.

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Zhu, Xiaoliang, and Yongbin Ge. "Adaptive High-Order Finite Difference Analysis of 2D Quenching-Type Convection-Reaction-Diffusion Equation." Advances in Mathematical Physics 2020 (October 29, 2020): 1–19. http://dx.doi.org/10.1155/2020/3650703.

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Quenching characteristics based on the two-dimensional (2D) nonlinear unsteady convection-reaction-diffusion equation are creatively researched. The study develops a 2D compact finite difference scheme constructed by using the first and the second central difference operator to approximate the first-order and the second-order spatial derivative, Taylor series expansion rule, and the reminder-correction method to approximate the three-order and the four-order spatial derivative, respectively, and the forward difference scheme to discretize temporal derivative, which brings the accuracy resulted meanwhile. Influences of degenerate parameter, convection parameter, and the length of the rectangle definition domain on quenching behaviors and performances of special quenching cases are discussed and evaluated by using the proposed scheme on the adaptive grid. It is feasible for the paper to offer potential support for further research on quenching problem.

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Frances Monllor, Jorge, Jani Tervo, and Cristian Neipp. "SPLIT-FIELD FINITE-DIFFERENCE TIME-DOMAIN SCHEME FOR KERR-TYPE NONLINEAR PERIODIC MEDIA." Progress In Electromagnetics Research 134 (2013): 559–79. http://dx.doi.org/10.2528/pier12101514.

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Mulder, W. A. "A simple finite-difference scheme for handling topography with the second-order wave equation." GEOPHYSICS 82, no. 3 (May 1, 2017): T111—T120. http://dx.doi.org/10.1190/geo2016-0212.1.

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ABSTRACT The presence of topography poses a challenge for seismic modeling with finite-difference codes. The representation of topography by means of an air layer or vacuum often leads to a substantial loss of numerical accuracy. A suitable modification of the finite-difference weights near the free surface can decrease that error. An existing approach requires extrapolation of interior solution values to the exterior while using the boundary condition at the free surface. However, schemes of this type occasionally become unstable and may be impossible to implement with highly irregular topography. One-dimensional extrapolation along coordinate lines results in a simple and efficient scheme. The stability of the 1D scheme is improved by ignoring the interior point nearest to the boundary during extrapolation in case its distance to the boundary is less than half a grid spacing. The generalization of the 1D scheme to more than one dimension requires a modification if the boundary intersects the finite-difference stencil on both sides of the central evaluation point and if there are not enough interior points to build the finite-difference stencil. Examples for the 2D constant-density acoustic case with a fourth-order finite-difference scheme demonstrate the method’s capability. Because the 1D assumption is not valid in two dimensions if the boundary does not follow grid lines, the formal numerical accuracy is not always obtained, but the method can handle highly irregular topography.

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Farooqi, Asma, Riaz Ahmad, Rashada Farooqi, Sayer O. Alharbi, Dumitru Baleanu, Muhammad Rafiq, Ilyas Khan, and M. O. Ahmad. "An Accurate Predictor-Corrector-Type Nonstandard Finite Difference Scheme for an SEIR Epidemic Model." Journal of Mathematics 2020 (December 15, 2020): 1–18. http://dx.doi.org/10.1155/2020/8830829.

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The present work deals with the construction, development, and analysis of a viable normalized predictor-corrector-type nonstandard finite difference scheme for the SEIR model concerning the transmission dynamics of measles. The proposed numerical scheme double refines the solution and gives realistic results even for large step sizes, thus making it economical when integrating over long time periods. Moreover, it is dynamically consistent with a continuous system and unconditionally convergent and preserves the positive behavior of the state variables involved in the system. Simulations are performed to guarantee the results, and its effectiveness is compared with well-known numerical methods such as Runge–Kutta (RK) and Euler method of a predictor-corrector type.

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Steppeler, J., P. Rípodas, B. Jonkheid, and S. Thomas. "Third-Order Finite-Difference Schemes on Icosahedral-Type Grids on the Sphere." Monthly Weather Review 136, no. 7 (July 1, 2008): 2683–98. http://dx.doi.org/10.1175/2007mwr2182.1.

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Abstract A practical method is proposed to achieve high-order finite-difference schemes on grids that are quasi-homogeneous on the sphere. A family of grids is used that are characterized by the parameter NP, which can take on values of 3, 4, and 5, etc. The parameter NP is the number of grid patches meeting at the Poles. For NP = 3 the cube sphere grid is obtained and for NP = 5 the icosahedron is obtained. While the grid construction method is valid for all values of NP, the tests performed in this paper concern only the case NP = 5 (i.e., the icosahedron). For each of the rhomboidal patches, the grid is created by connecting points on opposing sides of the rhomboid by great circles. This offers the possibility to obtain derivatives for a line of grid points along a great circle in the classical way. Therefore, it becomes possible to use well-known spatial discretizations from limited-area models. Local models can be transferred to the sphere with rather limited effort. The method was tested using the fourth-order Runge–Kutta integration method and fourth-order spatial differencing. At patch limits, boundary values are obtained using third-order serendipity interpolation, giving the scheme an overall space–time accuracy of 3. The serendipity interpolation is quite efficient. Third-order interpolation in two dimensions is achieved by a set of linear interpolations and a number of function evaluations. All coefficients can be precomputed. The third-order convergence is demonstrated by numerical experiments using Williamson’s test cases 2 and 6.

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Shishkin, G., L. Shishkina, and K. Cronin. "A NUMERICAL METHOD FOR A STEFAN-TYPE PROBLEM." Mathematical Modelling and Analysis 16, no. 1 (April 8, 2011): 119–42. http://dx.doi.org/10.3846/13926292.2011.562930.

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A Stefan-type problem is considered. This is an initial-boundary value problem on a composite domain for a parabolic reaction-diffusion equation with a moving interface boundary. At the moving boundary between the two subdomains, an interface condition is prescribed for the solution of the problem and its derivatives. A finite difference scheme is constructed that approximates the initial-boundary value problem. An iterative Newton-type method for the solution of the difference scheme and a numerical method for the analysis of the errors of the computed discrete solutions are both developed.

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Wang, Yinghua, Bao-Shan Wang, and Wai Sun Don. "Generalized Sensitivity Parameter Free Fifth Order WENO Finite Difference Scheme with Z-Type Weights." Journal of Scientific Computing 81, no. 3 (July 3, 2019): 1329–58. http://dx.doi.org/10.1007/s10915-019-00998-z.

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Wang, Guodong. "An Engquist–Osher type finite difference scheme with a discontinuous flux function in space." Journal of Computational and Applied Mathematics 235, no. 17 (July 2011): 4966–77. http://dx.doi.org/10.1016/j.cam.2011.04.024.

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Lu, Changna, Luoyan Xie, and Hongwei Yang. "The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography." Mathematical Problems in Engineering 2018 (2018): 1–15. http://dx.doi.org/10.1155/2018/2652367.

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A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.

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Liu, Weijie, Yue Ning, Fengyan Shi, and Zhilin Sun. "A 2DH fully dispersive and weakly nonlinear Boussinesq-type model based on a finite-volume and finite-difference TVD-type scheme." Ocean Modelling 147 (March 2020): 101559. http://dx.doi.org/10.1016/j.ocemod.2019.101559.

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Zhou, Ming, Chuan Peng Pan, L. P. Liu, R. Yuan, R. F. Ren, and Lan Cai. "Finite Difference Time Domain Method for Computing the Band-Structure of 3D Photonic Crystals." Solid State Phenomena 121-123 (March 2007): 599–602. http://dx.doi.org/10.4028/www.scientific.net/ssp.121-123.599.

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A numerical method based on Finite Difference Time Domain (FDTD) scheme for computing the photonic band-structure of three dimensional photonic crystals is introduced in this paper. Also, the accuracy and stability, numerical dispersion, boundary Conditions as well as excitation attaching to the scheme are detailed analyzed. For checking the method, the simulating results of photonic band structure on two type lattices are presented.

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Kalis, H. "THE EXACT FINITE-DIFFERENCE SCHEME FOR VECTOR BOUNDARY‐VALUE PROBLEMS WITH PIECE‐WISE CONSTANT COEFFICIENTS." Mathematical Modelling and Analysis 3, no. 1 (December 15, 1998): 114–23. http://dx.doi.org/10.3846/13926292.1998.9637094.

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We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schemes with perturbation coefficient of function‐matrix for solving the system of differential equations. Special finite‐difference approximations are constructed for a steady‐state boundary‐value problem, systems of parabolic type partial differential equations, a system of two MHD equations, 2‐D flows and MHD‐flows equations in curvilinear orthogonal coordinates.

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Lee, Seunggyu. "Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 2 (April 26, 2019): 137–43. http://dx.doi.org/10.1515/ijnsns-2017-0278.

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AbstractWe propose a fourth-order spatial and second-order temporal accurate and unconditionally stable compact finite-difference scheme for the Cahn–Hilliard equation. The proposed scheme has a higher-order accuracy in space than conventional central difference schemes even though both methods use a three-point stencil. Its compactness may be useful when applying the scheme to numerical implementation. In a temporal discretization, the secant-type algorithm, which is known as the second-order accurate scheme, is applied. Furthermore, the unique solvability regardless of the temporal and spatial step size, unconditionally gradient stability, and discrete mass conservation are proven. It guarantees that large temporal and spatial step sizes could be used with the high-order accuracy and the original properties of the CH equation. Then, numerical results are presented to confirm the efficiency and accuracy of the proposed scheme. The efficiency of the proposed scheme is better than other low order accurate stable schemes.

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ROUY, ELISABETH. "NUMERICAL APPROXIMATION OF VISCOSITY SOLUTIONS OF FIRST-ORDER HAMILTON-JACOBI EQUATIONS WITH NEUMANN TYPE BOUNDARY CONDITIONS." Mathematical Models and Methods in Applied Sciences 02, no. 03 (September 1992): 357–74. http://dx.doi.org/10.1142/s0218202592000223.

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We present a general result concerning numerical approximations, obtained by finite difference schemes, of viscosity solutions to the Cauchy problem for first-order Hamilton-Jacobi equations with Neumann type boundary conditions. It states that if Δt is the time-discretization , then the error estimate between the approximation and the solution is of the order [Formula: see text] under certain assumptions of monotonicity and consistency on the numerical scheme.

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Bommaraju, C., R. Marklein, and P. K. Chinta. "Optimally Accurate Second-Order Time-Domain Finite-Difference Scheme for Acoustic, Electromagnetic, and Elastic Wave Modeling." Advances in Radio Science 3 (May 12, 2005): 175–81. http://dx.doi.org/10.5194/ars-3-175-2005.

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Abstract. Numerical methods are extremely useful in solving real-life problems with complex materials and geometries. However, numerical methods in the time domain suffer from artificial numerical dispersion. Standard numerical techniques which are second-order in space and time, like the conventional Finite Difference 3-point (FD3) method, Finite-Difference Time-Domain (FDTD) method, and Finite Integration Technique (FIT) provide estimates of the error of discretized numerical operators rather than the error of the numerical solutions computed using these operators. Here optimally accurate time-domain FD operators which are second-order in time as well as in space are derived. Optimal accuracy means the greatest attainable accuracy for a particular type of scheme, e.g., second-order FD, for some particular grid spacing. The modified operators lead to an implicit scheme. Using the first order Born approximation, this implicit scheme is transformed into a two step explicit scheme, namely predictor-corrector scheme. The stability condition (maximum time step for a given spatial grid interval) for the various modified schemes is roughly equal to that for the corresponding conventional scheme. The modified FD scheme (FDM) attains reduction of numerical dispersion almost by a factor of 40 in 1-D case, compared to the FD3, FDTD, and FIT. The CPU time for the FDM scheme is twice of that required by the FD3 method. The simulated synthetic data for a 2-D P-SV (elastodynamics) problem computed using the modified scheme are 30 times more accurate than synthetics computed using a conventional scheme, at a cost of only 3.5 times as much CPU time. The FDM is of particular interest in the modeling of large scale (spatial dimension is more or equal to one thousand wave lengths or observation time interval is very high compared to reference time step) wave propagation and scattering problems, for instance, in ultrasonic antenna and synthetic scattering data modeling for Non-Destructive Testing (NDT) applications, where other standard numerical methods fail due to numerical dispersion effects. The possibility of extending this method to staggered grid approach is also discussed. The numerical FD3, FDTD, FIT, and FDM results are compared against analytical solutions.

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Mbroh, Nana Adjoah, Suares Clovis Oukouomi Noutchie, and Rodrigue Yves M’pika Massoukou. "A uniformly convergent finite difference scheme for Robin type singularly perturbed parabolic convection diffusion problem." Mathematics and Computers in Simulation 174 (August 2020): 218–32. http://dx.doi.org/10.1016/j.matcom.2020.03.003.

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Wegner, J. L., L. Jiang, and J. B. Haddow. "Application of a second-order Godunov-type finite difference scheme to a nonlinear elastodynamic problem." Computational Mechanics 8, no. 6 (1991): 355–63. http://dx.doi.org/10.1007/bf00370152.

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Xie, Chuang, Peng Song, Jun Tan, Baohua Liu, Jinshan Li, Kaiben Yu, Xiaobo Zhang, Chao Zhang, Hongyang Zhang, and Ruiqi Zhang. "Cosine-type weighted hybrid absorbing boundary based on the second-order Higdon boundary condition and its GPU implementation." Journal of Geophysics and Engineering 17, no. 2 (January 17, 2020): 231–48. http://dx.doi.org/10.1093/jge/gxz102.

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Abstract In this paper, we analyze the absorption effect of the hybrid absorbing boundary condition (ABC) with various hybrid modes, and propose a cosine-type optimized weighted hybrid mode taking into account the boundary reflection intensity of the inner and outer boundaries of transition area. Additionally, we derive a new finite-difference scheme of the second-order Higdon ABC and the corner equation for Graphic Processing Unit (GPU) acceleration. On this basis, a new type of second-order Higdon hybrid ABC applicable for GPU acceleration is established in the acoustic finite-difference modeling. Numerical experiments demonstrate that the proposed cosine-type weighted hybrid mode can achieve a better absorption effect compared with other weighted hybrid modes; the second-order Higdon ABC based on the proposed new finite-difference scheme can effectively improve the GPU speed-up ratio and is more effective in large-scale wavefield high-precision simulation based on GPU acceleration.

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Sadybekov, Makhmud A. "Stable difference scheme for a nonlocal boundary value heat conduction problem." e-Journal of Analysis and Applied Mathematics 1, no. 1 (December 1, 2018): 1–10. http://dx.doi.org/10.2478/ejaam-2018-0001.

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AbstractIn this paper, a new finite difference method to solve nonlocal boundary value problems for the heat equation is proposed. The most important feature of these problems is the non-self-adjointness. Because of the non-self-adjointness, major difficulties occur when applying analytical and numerical solution techniques. Moreover, problems with boundary conditions that do not possess strong regularity are less studied. The scope of the present paper is to justify possibility of building a stable difference scheme with weights for mentioned type of problems above.

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Parovik, Roman, and Dmitriy Tverdyi. "Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation." Mathematical and Computational Applications 26, no. 3 (July 29, 2021): 55. http://dx.doi.org/10.3390/mca26030055.

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The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-difference scheme is conditionally stable and converges to the first order. Using the fractional Riccati equation as an example, the computational accuracy of the numerical method is analyzed. It is shown that with an increase in the nodes of the computational grid, the order of computational accuracy tends to unity, i.e., to the theoretical value of the order of accuracy.

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Садовский, В. М., О. В. Садовская, and Е. А. Ефимов. "One-dimensional finite difference schemes for splitting method realization in axisymmetric equations of the dynamics of elastic medium." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 1 (February 3, 2021): 47–66. http://dx.doi.org/10.26089/nummet.v22r104.

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Строятся экономичные разностные схемы сквозного счета для решения прямых задач сейсмики в осесимметричной постановке. При распараллеливании алгоритмов, реализующих схемы на многопроцессорных вычислительных системах, применяется метод двуциклического расщепления по пространственным переменным. Одномерные системы уравнений на этапах расщепления решаются на основе явных сеточно-характеристических схем и неявной разностной схемы типа "предиктор-корректор" с контролируемой искусственной диссипацией энергии. Верификация алгоритмов и программ выполнена на точных решениях одномерных задач типа бегущих монохроматических волн. Сравнение результатов показало неоспоримые преимущества схемы с контролируемой диссипацией энергии по точности расчета гладких решений и целесообразность применения явных монотонных схем при расчете разрывов. We construct efficient finite difference shock-capturing schemes for the solution of direct seismic problems in axisymmetric formulation. When parallelizing the algorithms implementing the schemes on multiprocessor computing systems, the two-cyclic splitting method with respect to the spatial variables is used. One-dimensional systems of equations are solved at the stages of splitting on the basis of explicit gridcharacteristic schemes and an implicit finite difference scheme of the “predictor–corrector” type with controllable artificial energy dissipation. The verification of algorithms and programs is fulfilled on the exact solutions of one-dimensional problems describing traveling monochromatic waves. The comparison of the results showed the advantages of the scheme with controllable energy dissipation in terms of the accuracy of computing smooth solutions and the advisability of application of explicit monotone schemes when calculating discontinuities.

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Wang, Caihua. "A New Way to Generate an Exponential Finite Difference Scheme for 2D Convection-Diffusion Equations." Journal of Applied Mathematics 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/457938.

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The idea of direction changing and order reducing is proposed to generate an exponential difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion equation with source term. During the derivation process, the higher order derivatives alongy-direction are removed to the derivatives alongx-direction iteratively using information given by the original differential equation (similarly fromx-direction toy-direction) and then instead of keeping finite terms in the Taylor series expansion, infinite terms which constitute convergent series are kept on deriving the exponential coefficients of the scheme. From the construction process one may gain more insight into the relations among the stencil coefficients. The scheme is of positive type so it is unconditionally stable and the convergence rate is proved to be of second-order. Fourth-order accuracy can be obtained by applying Richardson extrapolation algorithm. Numerical results show that the scheme is accurate, stable, and especially suitable for convection-dominated problems with different kinds of boundary layers including elliptic and parabolic ones. The idea of the method can be applied to a wide variety of differential equations.

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Lee, S. L., and C. R. Ou. "Integration Scheme for Elastic Deformation and Stresses." Journal of Applied Mechanics 66, no. 4 (December 1, 1999): 978–85. http://dx.doi.org/10.1115/1.2791808.

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The integration scheme is proposed in this paper to solve linear deformation and stresses for elastic bodies. The discretized equations are of the finite difference type such that all of the advantages in the use of a finite difference scheme are preserved. In addition, the boundary traction can be easily converted into Dirichlet boundary condition for the displacement equations without recourse to fictitious points. Three examples are illustrated in this study to examine the performances of the integration scheme. In the case of thermal loading, the integration scheme is seen to provide solution with six-place accuracy while the finite element and the boundary element solutions possess only two- to three-place accuracy at essentially the same number of grid points. A similar situation is believed to exist also in the case of pure mechanical loading, although no exact solution is available for comparison. For a square bimaterial under a thermal loading without boundary traction, the integration scheme is found to successfully predict the existence of the interface zone. Due to its simplicity and efficiency, the integration scheme is expected to have good performance for solid mechanical problems, especially when coupled with heat transfer and fluid flow inside and outside the solid.

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FLOURI, EVANGELIA T., JOHN A. EKATERINARIS, and NIKOLAOS A. KAMPANIS. "HIGH-ORDER ACCURATE NUMERICAL SCHEMES FOR THE PARABOLIC EQUATION." Journal of Computational Acoustics 13, no. 04 (December 2005): 613–39. http://dx.doi.org/10.1142/s0218396x05002888.

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Efficient, high-order accurate methods for the numerical solution of the standard (narrow-angle) parabolic equation for underwater sound propagation are developed. Explicit and implicit numerical schemes, which are second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The explicit schemes have severe stability limitations and some of the proposed high-order accurate implicit methods were found conditionally stable. The efficiency and accuracy of various numerical methods are evaluated for Cartesian-type meshes. The standard parabolic equation is transformed to body fitted curvilinear coordinates. An unconditionally stable, implicit finite-difference scheme is used for numerical solutions in complex domains with deformed meshes. Simple boundary conditions are used and the accuracy of the numerical solutions is evaluated by comparing with an exact solution. Numerical solutions in complex domains obtained with a finite element method show excellent agreement with results obtained with the proposed finite difference methods.

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Benyo, Krisztian, Ayoub Charhabil, Mohamed-Ali Debyaoui, and Yohan Penel. "Simulation of complex free surface flows." ESAIM: Proceedings and Surveys 70 (2021): 45–67. http://dx.doi.org/10.1051/proc/202107004.

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We study the Serre-Green-Naghdi system under a non-hydrostatic formulation, modelling incompressible free surface flows in shallow water regimes. This system, unlike the well-known (nonlinear) Saint-Venant equations, takes into account the effects of the non-hydrostatic pressure term as well as dispersive phenomena. Two numerical schemes are designed, based on a finite volume - finite difference type splitting scheme and iterative correction algorithms. The methods are compared by means of simulations concerning the propagation of solitary wave solutions. The model is also assessed with experimental data concerning the Favre secondary wave experiments [12].

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de la Puente, Josep, Miguel Ferrer, Mauricio Hanzich, José E. Castillo, and José M. Cela. "Mimetic seismic wave modeling including topography on deformed staggered grids." GEOPHYSICS 79, no. 3 (May 1, 2014): T125—T141. http://dx.doi.org/10.1190/geo2013-0371.1.

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Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

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Don, Wai-Sun, Antonio de Gregorio, Jean-Piero Suarez, and Gustaaf B. Jacobs. "Assessing the Performance of a Three Dimensional Hybrid Central-WENO Finite Difference scheme with Computation of a Sonic Injector in Supersonic Cross Flow." Advances in Applied Mathematics and Mechanics 4, no. 06 (December 2012): 719–36. http://dx.doi.org/10.4208/aamm.12-12s03.

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AbstractA hybridization of a high order WENO-Zfinite difference scheme and a high order central finite difference method for computation of the two-dimensional Euler equations first presented in [B. Costa and W. S. Don, J. Comput. Appl. Math., 204(2) (2007)] is extended to three-dimensions and for parallel computation. The Hybrid scheme switches dynamically from a WENO-Zscheme to a central scheme at any grid location and time instance if the flow is sufficiently smooth and vice versa if the flow is exhibiting sharp shock-type phenomena. The smoothness of the flow is determined by a high order multi-resolution analysis. The method is tested on a benchmark sonic flow injection in supersonic cross flow. Increase of the order of the method reduces the numerical dissipation of the underlying schemes, which is shown to improve the resolution of small dynamic vortical scales. Shocks are captured sharply in an essentially non-oscillatory manner via the high order shock-capturing WENO-Zscheme. Computations of the injector flow with a WENO-Zscheme only and with the Hybrid scheme are in very close agreement. Thirty percent of grid points require a computationally expensive WENO-Zscheme for high-resolution capturing of shocks, whereas the remainder of grid points may be solved with the computationally more affordable central scheme. The computational cost of the Hybrid scheme can be up to a factor of one and a half lower as compared to computations with a WENO-Zscheme only for the sonic injector benchmark.

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CHRISTY ROJA, J., and A. TAMILSELVAN. "SHOOTING METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE." International Journal of Computational Methods 10, no. 06 (May 2, 2013): 1350041. http://dx.doi.org/10.1142/s0219876213500412.

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A class of singularly perturbed boundary value problems (SPBVPs) for fourth-order ordinary differential equations (ODEs) is considered. The SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. In order to solve them numerically, a method is suggested in which the given interval is divided into two inner regions (boundary layer regions) and one outer region. Two initial-value problems associated with inner regions and one boundary value problem corresponding to the outer region are derived from the given SPBVP. In each of the two inner regions, an initial value problem is solved by using fitted mesh finite difference (FMFD) scheme on Shishkin mesh and the boundary value problem corresponding to the outer region is solved by using classical finite difference (CFD) scheme on Shishkin mesh. A combination of the solution so obtained yields a numerical solution of the boundary value problem on the whole interval. First, in this method, we find the zeroth-order asymptotic expansion approximation of the solution of the weakly coupled system. Error estimates are derived. Examples are presented to illustrate the numerical method. This method is suitable for parallel computing.

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Yadav, Swati, Rajesh K. Pandey, Anil K. Shukla, and Kamlesh Kumar. "High-order approximation for generalized fractional derivative and its application." International Journal of Numerical Methods for Heat & Fluid Flow 29, no. 9 (September 2, 2019): 3515–34. http://dx.doi.org/10.1108/hff-11-2018-0700.

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Purpose This paper aims to present a high-order scheme to approximate generalized derivative of Caputo type for μ ∈ (0,1). The scheme is used to find the numerical solution of generalized fractional advection-diffusion equation define in terms of the generalized derivative. Design/methodology/approach The Taylor expansion and the finite difference method are used for achieving the high order of convergence which is numerically demonstrated. The stability of the scheme is proved with the help of Von Neumann analysis. Findings Generalization of fractional derivatives using scale function and weight function is useful in modeling of many complex phenomena occurring in particle transportation. The numerical scheme provided in this paper enlarges the possibility of solving such problems. Originality/value The Taylor expansion has not been used before for the approximation of generalized derivative. The order of convergence obtained in solving generalized fractional advection-diffusion equation using the proposed scheme is higher than that of the schemes introduced earlier.

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Meškauskas, T., and F. Ivanauskas. "Initial Boundary-Value Problems for Derivative Nonlinear Schroedinger Equation. Justification of Two-Step Algorithm." Nonlinear Analysis: Modelling and Control 7, no. 2 (December 5, 2002): 69–104. http://dx.doi.org/10.15388/na.2002.7.2.15195.

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We investigate two different initial boundary-value problems for derivative nonlinear Schrödinger equation. The boundary conditions are Dirichlet or generalized periodic ones. We propose a two-step algorithm for numerical solving of this problem. The method consists of Bäcklund type transformations and difference scheme. We prove the convergence and stability in C and H1 norms of Crank–Nicolson finite difference scheme for the transformed problem. There are no restrictions between space and time grid steps. For the derivative nonlinear Schrödinger equation, the proposed numerical algorithm converges and is stable in C1 norm.

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Amosov, A. A., and A. E. Vestfalsky. "Finite-difference scheme for two-scale homogenized equations of one-dimensional motion of a thermoviscoelastic Voigt-type body." Computational Mathematics and Mathematical Physics 46, no. 4 (April 2006): 691–718. http://dx.doi.org/10.1134/s0965542506040142.

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Hanafi, L., M. Mardlijah, D. B. Utomo, and A. Amiruddin. "Study numerical scheme of finite difference for solution partial differential equation of parabolic type to heat conduction problem." Journal of Physics: Conference Series 1821, no. 1 (March 1, 2021): 012032. http://dx.doi.org/10.1088/1742-6596/1821/1/012032.

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ZHANG, MENGPING, and CHI-WANG SHU. "AN ANALYSIS OF THREE DIFFERENT FORMULATIONS OF THE DISCONTINUOUS GALERKIN METHOD FOR DIFFUSION EQUATIONS." Mathematical Models and Methods in Applied Sciences 13, no. 03 (March 2003): 395–413. http://dx.doi.org/10.1142/s0218202503002568.

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In this paper we present an analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. The first formulation yields an numerically inconsistent and weakly unstable scheme, while the other two formulations, the local discontinuous Galerkin approach and the Baumann–Oden approach, give stable and convergent results. When written as finite difference schemes, such a distinction among the three formulations cannot be easily analyzed by the usual truncation errors, because of the phenomena of supraconvergence and weak instability. We perform a Fourier type analysis and compare the results with numerical experiments. The results of the Fourier type analysis agree well with the numerical results.

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Bernth, Henrik, and Chris Chapman. "A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids." GEOPHYSICS 76, no. 3 (May 2011): WA43—WA50. http://dx.doi.org/10.1190/1.3555530.

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Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, the dispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-difference calculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and four in three dimensions, which decouple in isotropic media. Therefore the Lebedev scheme will have twice or four times the computational requirements, memory, and CPU as the Virieux grid but can be used with general anisotropy. In two dimensions, the rotated staggered grid is exactly equivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numerical dispersion in finite-difference grids depends on the direction of propagation and the grid type and parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computational requirements for the two grid types, the dispersion, averaged over propagation direction and medium velocity are calculated. Setting the parameters so the average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field data and at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

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Lissy, Pierre, and Ionel Rovenţa. "Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method." Mathematical Models and Methods in Applied Sciences 30, no. 03 (March 2020): 439–75. http://dx.doi.org/10.1142/s0218202520500116.

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We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. The proof is mainly based on a (non-classic) moment method.

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Jha, Navnit, Venu Gopal, and Bhagat Singh. "Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 06 (December 2018): 1850053. http://dx.doi.org/10.1142/s1793962318500538.

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By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.

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Alpatov, Vadim. "Building constructions modelling problems in modern cae-systems." E3S Web of Conferences 135 (2019): 03066. http://dx.doi.org/10.1051/e3sconf/201913503066.

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This article is about choosing building structure analysis model question. Construction analysis model type choice affects result of counting. Result accuracy and reliability depends on analysis scheme choice. Using modern computers, there is a lot of alternative abilities of creation construction analysis model. This article is made to understand these analysis models features and their influence on result. An example of counting beam in six independent CAD systems is shown. Engineering simulation and design software, which base on finite elements method, were chosen for analysis. These counting models differ from each other only by geometrical scheme. Beam geometry modeling is performed using beam elements, shell elements and solid elements. The comparison of beam calculation results with its beam, shell and solid elements analysis scheme modelling was performed. The analysis of single factor (geometric scheme) influence on the results of beam calculation is shown. It was defined, that the choice of calculation complex does not affect the calculation result, if the geometrical counting models are completely identical. It was defined, that in case creating construction with various types finite elements there are differences in the calculation results. Difference in calculation results, using different geometrical models, is seen in using the same complex and in comparison of different complexes. It was defined, that difference in calculating internal forces and moments in beam for different geometrical models can be more than 10%.

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Journal articles on the topic 'Type-depedent finite difference scheme'

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