Heinz-Otto Kreiss

Stability theory of difference approximations for mixed initial boundary value problems. II

Stability theory of difference approximations for mixed initial boundary value problems. II

Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren

For difference equations with constant coefficients necessary and sufficient algebraic stability conditions are given for the stability definitions used by G. Forsythe and W. Wasow (A) and P. D. Lax and R. D. Richtmyer (B). The application of these conditions for difference equations with variable coefficients is considered and it is shown that the stability condition of definitionA is not sufficient for stability. The same is true with respect to the definitionB if the difference equations are not parabolic and do not approximate first order systems. Therefore another stability definition is proposed and a number of properties are discussed.

Stability of the Fourier method

In this paper we develop a stability theory for the Fourier (or pseudo-spectral) method for linear hyperbolic and parabolic partial differential equations with variable coefficients.

Difference approximations for singular perturbations of systems of ordinary differential equations

SummaryWe consider difference methods for the solution of singular perturbations of boundary value problems. The solutions are smooth except in boundary layers of thickness ε|logε|, 0<ε≪1. Various difference schemes with a uniform stepsizeh are considered. In practice,h is usually much larger than the boundary layer regions. Then the difference methods must be choosen with care. It is shown that only approximations of low order accuracy can be used. However, one can increase the accuracy by a Richardson procedure. Asymptotic expansions in powers ofh and ε are given for the solutions of the proposed methods.

Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations

We study the slightly compressible Navier-Stokes equations. We first consider the Cauchy problem, periodic in space. Under appropriate assumptions on the initial data, the solution of the compressible equations consists-to first order-of a solution of the incompressible equations plus a function which is highly oscillatory in time. We show that the highly oscillatory part (the sound waves) can be described by wave equations, at least locally in time. We also show that the bounded derivative principle is valid; i.e., the highly oscillatory part can be suppressed by initialization. Besides the Cauchy problem, we also consider an initial-boundary value problem. At the inflow boundary, the viscous term in the Navier-Stokes equations is important. We consider the case where the compressible pressure is prescribed at inflow. In general, one obtains a boundary layer in the pressure; in the velocities a boundary layer is not present to first approximation.

Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation

We consider the three-dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress-free boundary condition. Building on our recently developed theory for difference methods for second order hyperbolic systems [H.-O. Kreiss, N. A. Petersson, J. Ystrom, SIAM J. Numer. Anal., 40 (2002), pp. 1940-1967], we develop an explicit, second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities. The spatial discretization is self-adjoint, and the stability is obtained through an energy estimate. Seismic events are often modeled using singular source terms, and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source. Several numerical examples are given.

Difference Methods for Stiff Ordinary Differential Equations

Consider the initial value problem for a first order system of stiff ordinary differential equations. The smoothness properties of its solutions are investigated and a general theory for difference...

A Second Order Accurate Embedded Boundary Method for the Wave Equation with Dirichlet Data

The accuracy of Cartesian embedded boundary methods for the second order wave equation in general two-dimensional domains subject to Dirichlet boundary conditions is analyzed. Based on the analysis, we develop a numerical method where both the solution and its gradient are second order accurate. We avoid the small-cell stiffness problem without sacrificing the second order accuracy by adding a small artificial term to the Dirichlet boundary condition. Long-time stability of the method is obtained by adding a small fourth order dissipative term. Several numerical examples are provided to demonstrate the accuracy and stability of the method. The method is also used to solve the two-dimensional TM

Difference Approximations of the Neumann Problem for the Second Order Wave Equation

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Numerical methods for stiff two-point boundary value problems

problem for Maxwells equations posed as a second order wave equation for the electric field coupled to ordinary differential equations for the magnetic field.