ESERK Methods to Numerically Solve Nonlinear Parabolic PDEs in Complex Geometries: Using Right Triangles

In this paper Extrapolated Stabilized Explicit Runge-Kutta methods (ESERK) are proposed to solve nonlinear partial differential equations (PDEs) in right triangles. These algorithms evaluate more times the function than a standard explicit Runge--Kutta scheme ($n_t$ times per step), and these extra evaluations do not increase the order of convergence but the stability region grows with $\mathcal{O}(n_t^2)$. Hence, the total computational cost is $\mathcal{O}(n_t)$ times lower than with a traditional explicit algorithm. Thus, these algorithms have been traditionally considered to solve stiff PDEs in squares/rectangles or cubes. In this paper, for the first time, ESERK methods are considered in a right triangle. It is demonstrated that such type of codes keep the convergence and the stability properties under certain conditions. This new approach would allow to solve nonlinear parabolic PDEs with stabilized explicit Runge--Kutta schemes in complex domains, that would be decomposed in rectangles and right triangles.