Galerkin/Runge–Kutta Discretizations for Semilinear Parabolic Equations

A new class of fully discrete Galerkin/Runge–Kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems. Unlike any classical counterpart, this class offers arbitrarily high-order convergence without suffering from what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Furthermore, it is noted that special Runge–Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low-order method.