ESERK5: A fifth-order extrapolated stabilized explicit Runge–Kutta method

Abstract

A new algorithm is developed and analyzed for multi-dimensional non-linear parabolic partial differential equations (PDEs) which are semi-discretized in the spatial variables leading to a system of ordinary differential equations (ODEs). It is based on fifth-order extrapolated stabilized explicit Runge-Kutta schemes (ESERK). They are explicit methods, and therefore it is not necessary to employ complicated software for linear or non-linear system of equations. Additionally, they have extended stability regions along the negative real semi-axis, hence they can be considered to solve stiff problems coming from very common diffusion or reaction-diffusion problems.Previously, only lower-order codes (up to fourth-order) have been constructed and made available in the scientific literature. However, at the same time, higher-order codes were demonstrated to be very efficient to solve equations where it is necessary to have a high precision or they have transient zones that are very severe, and where functions change very fast. The new schemes allow changing the step length very easily and with a very small computational cost. Thus, a variable step length, with variable number of stages algorithm is constructed and compared with good numerical results in relation to other well-known ODE solvers.

A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting

“…For such problems, a few pre-smoothing steps involving a lower order scheme with an L-acceptable rational approximation to the exponential was recommended in [16]. We investigate presmoothing the ETDRK4P22-IF scheme with the third order ETDRK scheme proposed in [16], which approximates the matrix exponentials in ( 9)- (10) with Padé (0,3) rational functions. Note that this smoothing scheme is not implemented with dimensional splitting.…”

“…Among the host of time stepping methods available to solve stiff ODE systems [7,8,9,10,11] are the class of exponential time differencing (ETD)…”

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

“…These schemes typically have order 2 or 4 [8,[10][11][12][13][14][15][16]. Recently, we propose a new procedure combined with Richardson extrapolation to obtain methods with other orders of convergence [17,18], but in all these methods, these integrators have many more stages than the order of convergence. Most of these extra stages seek to extend as much as possible the region of stability along the negative real axis.…”