SERK2v3: Solving mildly stiff nonlinear partial differential equations

摘要

Multidimensional nonlinear parabolic partial differential equations (PDEs) appear in a large variety of disciplines. Usually, the scientific literature advises against the use of explicit ODE solvers for the integration of stiff problems. However, in this manuscript, we would like to show how some explicit methods can be very efficient for specific problems. Stabilized Explicit Runge–Kutta methods (SERK) are a class of explicit methods with extended stability domains along the negative real axis. It is necessary to evaluate the function s times, but the stability region is O(s2). Hence, the computational cost is O(s) times lower than for traditional explicit algorithms. In this way moderately stiff problems can be integrated by the use of simple explicit evaluations. SERK schemes can easily be applied to many different classes of problems with large dimensions and additionally they have low memory demand. Since these methods are explicit, they do not require algebra routines to solve large nonlinear systems associated to ODEs and therefore are especially well-suited for the method of lines (MOL) discretizations of parabolic multi-dimensional PDEs.Additionally, the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way, that is with minimal number of additional stages; hence, adaptation of the length step is possible at practically no extra cost. In this work, we study several strategies to change the time step for non-smooth functions and finally propose a new technique. This procedure improves the numerical results, especially when the data is non-smooth. We also derive two algorithms to estimate the spectral radius in the new code: a nonlinear power method and another procedure based on the Gershgorin theorem. Although SERK algorithms are only second-order schemes, in this manuscript we propose to utilize them after higher-order discretizations in space to obtain higher-accurate solutions efficiently and usually faster (since these higher-order schemes allow the use of a smaller number of nodes). Some numerical experiments shown in the paper support these conclusions.