Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations (Communications in Computational Physics, 2018)
It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods. More precisely,we can use spectral methods to postprocess numerical solutions of initial value differential equations. After a few number of iterations (say 3 to 4), the errors can decreaseto a few orders of magnitude less. The iteration uses the Gauss-Seidel type strategy,which gives an explicit way of postprocessing. Numerical examples for ODEs, Hamiltonian system and integral equations are provided. They all indicate that the spectralprocessing technique can be a very useful way in improving the accuracy of the numerical solutions. In particular, for a Hamiltonian system accuracy is only one of theissues; some other conservative properties are even more important for large time simulations. The spectral postprocessing with the coarse-mesh symplectic initial guess cannot only produce high accurate approximations but can also save a significant amountof computational time over the standard symplectic schemes.