Liouville–Green–Olver Approximations for Complex Difference Equations

Difference Approximations to Partial Differential Equations

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...

Finite Difference Approximations for Nonlinear First Order Partial Differential Equations

Classical solutions of nonlinear partial differential equations are approximated in the paper by solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on a comparison method. This new approach to the numerical solving of nonlinear equations is generated by a linea...

Compact Finite Difference Approximations for Space Fractional Diffusion Equations

Based on the weighted and shifted Grünwald difference (WSGD) operators [24], we further construct the compact finite difference discretizations for the fractional operators. Then the discretization schemes are used to approximate the one and two dimensional space fractional diffusion equations. The detailed numerical stability and error analysis are theoretically performed. We theoretically pro...

High Order Difference Approximations for the Linearized Euler Equations

The computers of today make it possible to do direct simulation of aeroacoustics, which is very computationally demanding since a very high resolution is needed. In the present thesis we study issues of relevance for aeroacoustic simulations. Paper A considers standard high order difference methods. We study two different ways of applying boundary conditions in a stable way. Numerical experimen...