Time-Dependent Problems and Difference Methods

@inproceedings{Gustafsson1996TimeDependentPA,
  title={Time-Dependent Problems and Difference Methods},
  author={Bertil Gustafsson and Heinz-Otto Kreiss and Joseph E. Oliger},
  year={1996}
}
Finite Difference Methods for 1st Order in Time, 2nd Order in Space Hyperbolic Systems used in Numerical Relativity
This thesis is concerned with the development of numerical methods using finite difference techniques for the discretization of initial value problems (IVPs) and initial boundary value problems
Numerical Methods for Evolutionary Differential Equations
TLDR
Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method useful.
A MATRIX-FREE LEGENDRE SPECTRAL METHOD FOR INITIAL-BOUNDARY VALUE PROBLEMS
TLDR
In this thesis it is proved that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.
A Sixth Order Accuracy Solution to a System of Nonlinear Differential Equations with Coupled Compact Method
TLDR
A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations and solved alternately and explicitly in time without linearizing the nonlinearity.
AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION
These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution
The Convergence of Spectral and Finite Difference Methods for Initial-Boundary Value Problems
TLDR
This paper proves that the Fourier method converges quadratically in the neighborhood of t=0 and the boundaries and quartically for large t when the first-order compatibility conditions are violated, and proves that an O(N-2log N) convergence is possible.
Stable and High-Order Finite Difference Methods for Multiphysics Flow Problems
TLDR
This thesis uses the finite difference method on summation-by-parts (SBP) form together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT) to provide a technique for overcoming most of the drawbacks of the finite Difference method.
Stable, Accurate and Efficient Interface Procedures for Viscous Problems
TLDR
It is shown that the hybrid method is an accurate, efficient and practically useful computational tool that can handle complex geometries and wave propagation phenomena.
...
...