Spectral Deferred Corrections for Parabolic Partial Differential Equations
Yale University New Haven United States
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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 On operations where n is the number of spatial discretization nodes.