Efficient inequality-preserving integrators for differential equations satisfying forward Euler conditions

Abstract

Developing explicit, high-order accurate, and stable algorithms for nonlinear differential equations remains an exceedingly difficult task. In this work, a systematic approach is proposed to develop high-order, large time-stepping schemes that can preserve inequality structures shared by a class of differential equations satisfying forward Euler conditions. Strong-stability-preserving (SSP) methods are popular and effective for solving equations of this type. However, few methods can deal with the situation when the time-step size is larger than that allowed by SSP methods. By adopting time-step-dependent stabilization and taking advantage of integrating factor methods in the Shu-Osher form, we propose enforcing the inequality structure preservation by approximating the exponential function using a novel recurrent approximation without harming the convergence. We define sufficient conditions for the obtained parametric Runge-Kutta (pRK) schemes to preserve inequality structures for any time-step size, namely, the underlying Shu-Osher coefficients are non-negative. To remove the requirement of a large stabilization term caused by stiff linear operators, we further develop inequality-preserving parametric integrating factor Runge-Kutta (pIFRK) schemes by incorporating the pRK with an integrating factor related to the stiff term, and enforcing the non-decreasing of abscissas. The only free parameter can be determined a priori based on the SSP coefficient, the time-step size, and the forward Euler condition. We demonstrate that the parametric methods developed here offer an effective and unified approach to study problems that satisfy forward Euler conditions, and cover a wide range of well-known models. Finally, numerical experiments reflect the high-order accuracy, efficiency, and inequality-preserving properties of the proposed schemes.