Spectral Deferred Corrections for Parabolic Partial Differential Equations

tions (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 stiff 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. However, in R2, the straightforward spatial discretization leads to matrices with dimensionality n2 × n2 and bandwidth n. While fast inversion schemes of O(n3) exist, we use alternating direction implicit (ADI) methods to replace the single two-dimensional implicit step with two sub-steps where only one direction is treated implicitly. This approach results in schemes with computational cost O(n2). Likewise, ADI methods in R3 have computational cost O(n3). While popular ADI methods are low-order, we combine the SDC methods with an ADI method to generate computationally efficient, high-order schemes for the solution of parabolic PDEs in R2 and R3. Third, traditional pseudospectral schemes for the representation of the spatial operator in parabolic PDEs yield differentiation operators with eigenvalues that can be excessively large. We improve on the traditional approach by subdividing the entire spatial domain, constructing bases on each subdomain, and combining the obtained discretization with the implicit SDC schemes. The resulting schemes are high-order in both time and space and have computational cost O(N ·M) where N is the number of spatial discretization nodes and M is the number of temporal nodes. We illustrate the behavior of these schemes with several numerical examples.