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Derivation of boundary conditions for the artificial boundaries associated with the solution of certain time dependent problems by Lax–Wendroff type difference schemes

John C. Wilson

1982
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Proceedings of the Edinburgh Mathematical Society
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Introduction Many problems involving the solution of partial differential equations require the solution over a finite region with fixed boundaries on which conditions are prescribed. It is a well known fact that the numerical solution of many such problems requires additional conditions on these boundaries and these conditions must be chosen to ensure stability. This problem has been considered by, amongst others, Kreiss [11, 12, 13] , Osher [16, 17], Gustafsson et al. [9] Gottlieb and Tarkel
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... ottlieb and Tarkel [7] and Burns [1] It is also necessary to solve partial differential equations over infinite domains, this type of problem occurring in transonic flows, seismology and meteorology. A numerical solution must be over a finite domain and one method of limiting the area of computation is to use artificial boundaries on which suitable conditions must be obtained. These conditions must be such that the solution of the modified problem is close to that of the original one over their common domain. If there is exact correspondence between the solutions the boundary condition becomes non-reflecting. Engquist and Majda [2, 3] examined such a problem, using the theory of pseudodifferential operators to construct well-posed boundary conditions for wave and other differential equations. Although the ideal non-reflecting boundary conditions are non-local in both time and space, practical computing considerations required them to use conditions which are local in both time and space. For the numerical problem these local conditions were approximated in a stable manner and it was shown that the resulting reflection at the artificial boundaries was small. In the second paper Engquist and Majda also considered the construction of radiation boundary conditions for the difference equation approximating the differential equation. For a finite difference approximation to the wave equation they obtained the symbol of the theoretical discrete-radiation boundary condition from the symbol of the approximation to the differential equation. Practical stable finite difference boundary conditions involving a parameter were given such that their symbol approximates closely that of the theoretical boundary condition. It is suggested that the parameter be chosen to minimise the truncation error of the boundary approximation and an estimate of the reflection coefficient for the boundary is given. Gustafsson and Kreiss [10] examine the problem of artificial boundaries in a different manner though some of the boundary approximations they derive are equivalent to the

doi:10.1017/s0013091500004053
fatcat:d3sxp4fnzfguphg24yaf3rsvme