| Numerical schemes often develop inaccuracies when used to solve parabolic partial differential equations with nonsinooth data e.g., from high frequency components. For homogeneous parabolic PDEs with nonsinooth data, Luskin and Rannacher [60, 61] discovered for the case of Crank-Nicolson, and Rannacher [69, 70] for the general diagonal Pade schemes, that use of certain steps of first subdiagonal Pade schemes at the start significantly recover the performance of diagonal Pade schemes.; Wade, Khaliq, Siddique, and Yousuf [91] have recently discovered a new class of higher order smoothing schemes for Homogeneous parabolic PDEs with nonsmooth data. These schemes are more robust in the sense of stability and accuracy than the well known Rannacher schemes. The new family of algorithms utilizes diagonal Pade schemes combined with the positivity preserving Pade schemes instead of first subdiagonal Pade schemes. Optimal order convergence for nonsmooth data, is proved in [91] for the case of a self-adjoint operator in Hilbert space.; The smoothing idea is extended to develop a family of schemes for inhomogeneous linear parabolic PDEs with nonsmooth data. Partial fraction decomposition is used to develop parallel algorithms to implement higher order methods in essentially the same time as that of a lower order method. These algorithms are tested on a number of one-dimensional and two-dimensional problems with constant and variable coefficients subject to constant as well as time dependent boundary conditions. Convergence is proved for problems with nonsmooth data for the case of a self-adjoint operator in Hilbert space.; Cox & Matthews [19] have recently developed a class of ETDRK (exponential time differencing Runge-Kutta) schemes for nonlinear, stiff systems of ODEs. Kassam and Trefethen [43] have shown that these schemes suffer from numerical instability. Using Pade schemes, we have developed a fourth-order ETDRK4 smoothing scheme to solve semilinear parabolic equations with nonsmooth data. The new scheme not only solve the problems of numerical instability but also computationally it is much more efficient than the original Cox & Matthews scheme. Numerical experiments support the analysis; including for the Allen-Cahn equation and a problem from Biochemistry. |
