Fully discrete hyperbolic initial boundary value problems with nonzero initial data

The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in \^a 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b]

Stability of finite difference schemes for hyperbolic initial boundary value problems

International audienceWe study the stability of finite difference schemes for hyperbolic initial boundary value problems in one space dimension. Assuming stability for the dicretization of the hyperbolic operator as well as a geometric regularity condition, we show that an appropriate determinant condition, that is the analogue of the uniform Kreiss-Lopatinskii condition for the continuous problem, yields strong stability for the discretized initial boundary value problem. The analysis relies on a suitable discrete block structure condition and the construction of suitable symmetrizers. Our work extends the results of Gustafsson, Kreiss, Sundstrom to a wider class of finite difference schemes

Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems

New convenient stability criteria are provided in this paper for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = au sub x + Bu + f in the quarter plane x or = 0, t or = 0. Using the new criteria, stability is easily established for numerous combinations of well known basic schemes and boundary conditions, thus generalizing many special cases studied in recent literature

Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation

Hyperbolic partial differential equations are frequently referenced in modeling real-world problems in mathematics and engineering. Therefore, in this study, an initial-boundary value issue is proposed for the pseudo-hyperbolic telegraph equation. By operator method, converting the PDE to an ODE provides an exact answer to this problem. After that, the finite difference method is applied to construct first-order finite difference schemes to calculate approximate numerical solutions. The stability estimations of finite difference schemes are shown, as well as some numerical tests to check the correctness in comparison to the precise solution. The numerical solution is subjected to error analysis. As a result of the error analysis, the maximum norm errors tend to decrease as we increase the grid points. It can be drawn that the established scheme is accurate and effectiv

Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.Comment: 20 page