Compact High Order Accurate Schemes for the Three Dimensional Wave Equation

Smith, Fouche; Tsynkov, Semyon; Turkel, Eli

doi:10.1007/s10915-019-00970-x

Cited by 27 publications

(20 citation statements)

References 24 publications

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“…Compact higher-order finite-difference schemes for PDEs is a popular subject and a vast literature is devoted to them. The case of such type schemes for the wave equation have recently attracted a lot of interest, in particular, see [2,4,8,12], where much more related references can be found.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, enlarging of most schemes to the case of the wave equation with the variable coefficient ρ(x) in front of ∂ 2 t u is simple, and there exists some connection to [2,12]. Also the main schemes are rather easily generalized for non-uniform rectangular meshes in space and time; we apply averaging technique to implement both aims.…”

Section: Introductionmentioning

confidence: 99%

“…For a i and h i independent on i, the formulas are simplified, and the operators on the left there differ only up to factors from ones appearing in the related formulas ( 21)-( 22) in [2] and (11) in [12]. Moreover, one can show that in this case generalized equations (3.22) for n = 2 and (3.26) for n = 3 are equivalent to respective methods from [2,12] up to approximations of f . But the stability and implementation issues in the generalized case are more complicated and are beyond the scope of this paper.…”

Section: Introductionmentioning

confidence: 99%

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On Compact 4th Order Finite-Difference Schemes for the Wave Equation

Злотник

Kireeva

2021

Mathematical Modelling and Analysis

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We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Compact 4th Order Finite-Difference Schemes for the Wave Equation

Злотник

Kireeva

2021

Mathematical Modelling and Analysis

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“…Such compact schemes are implicit and most often conditionally stable. Among them, the first type of schemes is constituted by implicit schemes which require application of FFT (for constant coefficients) or iterative methods (for variable coefficients) for their efficient implementation, see, in particular, [2,8,18,22,24], where additional references are contained.…”

Section: Introductionmentioning

confidence: 99%

On properties of an explicit in time fourth-order vector compact scheme for the multidimensional wave equation

Zlotnik

2021

Preprint

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An initial-boundary value problem for the n-dimensional wave equation is considered. A threelevel explicit in time and conditionally stable 4th-order compact scheme constructed recently for n = 2 and the square mesh is generalized to the case of any n 1 and the rectangular uniform mesh. Another approach to approximate the solution at the first time level (not exploiting high-order derivatives of the initial functions) is suggested. New stability bounds in the mesh energy norms and the discrete energy conservation laws are given, and the 4th order error bound is rigorously proved. Generalizations to the cases of the non-uniform meshes in space and time as well as of the wave equation with variable coefficients are suggested.

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“…A second order accurate scheme to deal with flux boundary conditions would ruin global fourth order accuracy if a fourth order discretization is utilized in the interior. Research on HOC schemes for flux type boundary conditions can be found, for example, in [4,6,20] for time dependent problems based on operator splitting approaches, for Poisson and Helmholtz or wave equations [1,5,11,16,22,23], other related HOC methods and applications [3,9,10,15,18,20,21,26,27], and for diffusion and advection equations [1,7,14]. However, few can be found in the literature for general elliptic partial differential equations with flux boundary conditions.…”

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confidence: 99%

High order compact schemes for flux type BCs

Li¹,

Pan²

2021

Preprint

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In this paper new innovative fourth order compact schemes for Robin and Neumann boundary conditions have been developed for boundary value problems of elliptic PDEs in two and three dimensions. Different from traditional finite difference operator approach, which may not work for flux type of boundary conditions, carefully designed undetermined coefficient methods are utilized in developing high order compact (HOC) schemes. The new methods not only can be utilized to design HOC schemes for flux type of boundary conditions but can also be applied to general elliptic PDEs including Poisson, Helmholtz, diffusion-advection, and anisotropic equations with linear boundary conditions. In the new developed HOC methods, the coefficient matrices are generally M-matrices, which guarantee the discrete maximum principle for well-posed problems, so the convergence of the HOC methods. The developed HOC methods are versatile and can cover most of high order compact schemes in the literature. The HOC methods for Robin boundary conditions and for anisotropic diffusion and advection equations with Robin or even Dirichlet boundary conditions are likely the first ones that have ever been developed. With the help of pseudo-inverse, or SVD solutions, we have also observed that the developed HOC methods usually have smaller error constants compared with traditional HOC methods when applicable. Non-trivial examples with large wave numbers and oscillatory solutions are presented to confirm the performance of the new HOC methods.

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Compact High Order Accurate Schemes for the Three Dimensional Wave Equation

Cited by 27 publications

References 24 publications

On Compact 4th Order Finite-Difference Schemes for the Wave Equation

On Compact 4th Order Finite-Difference Schemes for the Wave Equation

On properties of an explicit in time fourth-order vector compact scheme for the multidimensional wave equation

High order compact schemes for flux type BCs

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