Spectral Deferred Corrections for Parabolic Partial Differential Equations
Abstract
We describe a new class of algorithms for the solution of parabolic partial differential equations (PDEs). This class of schemes is based on three principal observations. First, the spatial discretization of parabolic PDEs results in stiff systems of ordinary differential equations (ODEs) in time, and therefore, requires an implicit method for its solution. Spectral Deferred Correction (SDC) methods use repeated iterations of a low-order method (e.g. implicit Euler method) to generate a high-order scheme. As a result, SDC methods of arbitrary order can be constructed with the desired stability properties necessary for the solution of sti differential equations. Furthermore, for large-scale systems, SDC methods are more computationally efficient than implicit Runge-Kutta schemes. Second, implicit methods for the solution of a system of linear ODEs yield linear systems that must be solved on each iteration. It is well known that the linear systems constructed from the spatial discretization of parabolic PDEs are sparse. In R1, these linear systems can be solved in O(n) operations where n is the number of spatial discretization nodes.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 08, 2015
- Accession Number
- AD1007861
Entities
People
- Daniel Berylkin
Organizations
- Yale University