A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound
We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of t...
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| Vydáno v: | Journal of scientific computing Ročník 76; číslo 2; s. 777 - 811 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York
Springer US
01.08.2018
Springer Nature B.V |
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| ISSN: | 0885-7474, 1573-7691 |
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| Abstract | We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems. |
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| AbstractList | We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems. |
| Author | Britt, Steven Tsynkov, Semyon Turkel, Eli |
| Author_xml | – sequence: 1 givenname: Steven orcidid: 0000-0003-4155-9401 surname: Britt fullname: Britt, Steven email: darrellstevenbritt@gmail.com organization: School of Mathematical Sciences, Tel Aviv University – sequence: 2 givenname: Eli surname: Turkel fullname: Turkel, Eli organization: School of Mathematical Sciences, Tel Aviv University – sequence: 3 givenname: Semyon surname: Tsynkov fullname: Tsynkov, Semyon organization: Department of Mathematics, North Carolina State University, Moscow Institute of Physics and Technology |
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| Keywords | Compact finite differences Wave equation Variable coefficients High order accuracy |
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| SubjectTerms | Accuracy Acoustics Algorithms Approximation Boundary conditions Boundary value problems Computational Mathematics and Numerical Analysis Conjugate gradient method Convergence Efficiency Eigenvalues Finite difference method Helmholtz equations Iterative methods Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Multigrid methods Theoretical Time dependence Time marching Upper bounds Wave equations |
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| Title | A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound |
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