Abstract
When using a finite difference method to solve a time dependent partial differential equation, the truncation error is often larger at a few grid points near a boundary or grid interface than in the interior. In computations, the observed convergence rate is often higher than the order of the large truncation error. In this paper, we develop techniques for analyzing this phenomenon, and particularly consider the second order wave equation. The equation is discretized by a finite difference operator satisfying a summation by parts property, and the boundary and grid interface conditions are imposed weakly by the simultaneous approximation term method. It is well-known that if the semi-discretized wave equation satisfies the determinant condition, that is the boundary system in Laplace space is nonsingular for all Re(s)ge 0, two orders are gained from the large truncation error localized at a few grid points. By performing a normal mode analysis, we show that many common discretizations do not satisfy the determinant condition at s=0. We then carefully analyze the error equation to determine the gain in the convergence rate. The result shows that stability does not automatically imply a gain of two orders in the convergence rate. The precise gain can be lower than, equal to or higher than two orders, depending on the boundary condition and numerical boundary treatment. The accuracy analysis is verified by numerical experiments, and very good agreement is obtained.
Highlights
In many physical problems arising in for example acoustics, seismology, and electromagnetism, the governing equations can be formulated as systems of second order time dependent hyperbolic partial differential equations (PDE)
For the second order wave equation a stable numerical scheme does not automatically satisfy the determinant condition, nor does it imply an optimal gain in convergence rate
We have considered stable summation-by-parts simultaneous–approximation–term (SBP–SAT) finite difference schemes for the Dirichlet, Neumann and interface problems
Summary
In many physical problems arising in for example acoustics, seismology, and electromagnetism, the governing equations can be formulated as systems of second order time dependent hyperbolic partial differential equations (PDE). One major difficulty with high order spatial discretizations is the numerical treatment of boundary conditions and grid interface conditions To achieve both stability and high accuracy, one candidate is the summation-by-parts simultaneous–approximation–term (SBP–SAT) finite difference method [4,22]. The analysis of gain in convergence for different PDEs has been a long–standing research topic It is well–known that by directly applying the energy method to the error equation, 1/2 order is gained in the convergence rate compared with the largest truncation error. Analysis in Laplace space is performed and yields sharper error estimates than the 1/2 order gain obtained by applying the energy method to the error equation in physical space.
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