Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

Stability Analysis of Numerical Boundary Conditions and Implicit Difference Approximations for Hyperbolic Equations

High-order finite difference approximations for hyperbolic problems: multiple penalties and non-reflecting boundary conditions

In this thesis, we use finite difference operators with the Summation-By-Parts property (SBP) and a weak boundary treatment, known as Simultaneous Approximation Terms (SAT), to construct high-order accurate numerical schemes. The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic probl...

Numerical stability for finite difference approximations of Einstein's equations

We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We pro...

Analytic solutions for the Stephen's inverse problem with local boundary conditions including Elliptic and hyperbolic equations

In this paper, two inverse problems of Stephen kind with local (Dirichlet) boundary conditions are investigated. In the first problem only a part of boundary is unknown and in the second problem, the whole of boundary is unknown. For the both of problems, at first, analytic expressions for unknown boundary are presented, then by using these analytic expressions for unknown boundaries and bounda...

A Priori Estimates for Mixed Finite Element Approximations of Second Order Hyperbolic Equations with Absorbing Boundary Conditions

A priori estimates for mixed nite element methods for the wave equations, 6] T. Dupont, L 2-estimates for Galerkin methods for second order hyperbolic equations, SIAM J.