Back to Departmental Colloquium: Spring 2006
Departmental Colloquium
Date: Thursday, Jan 12, 2006
Time: 4:15PM
Location: JWB 335
Natasha Flyer
National Center for Atmospheric Research
Title |
On the nature of initial-boundary value solutions and their ramifications for high-order methods |
Abstract |
It is obvious that if the initial (IC) and boundary condition (BC) do not agree in the corner of the time-space domain for an initial boundary value problem (IBVP), a discontinuity will arise in the solution. What is not so obvious is that in order for the solution to be $C^\infty$, the BC and IC must satisfy the partial differential equation (PDE) and all differentiated forms of it, forming an infinite set of compatibility conditions. The IC, being independent of the BC (otherwise one could pose an initial value problem instead), cannot satisfy both the boundary equation and all the compatibility conditions determined by the PDE on the boundary. As a result, a singularity will arise in some derivative of the solution. Although, the theory of compatibility conditions and the regularity of solutions for IBVPs is well known in the realm of theoretical mathematics, it has essentially gone unnoticed in the numerical community. Yet, its ramifications on the performance of high-order methods are severe. Here, we discuss the nature of IBV solutions for dissipative and dispersive equations and its impact on the convergence and accuracy of high-order methods. Examples will include the heat, linear KDV and Schr"odinger equations. |
Differential Equations
Computational Mathematics
Applied Mathematics