Unconditionally $L^2$-stable explicit--implicit--null filtered Runge--Kutta methods for general nonlinear degenerate parabolic equations

Abstract

We develop unconditionally $L^2$-stable explicit--implicit--null (EIN) filtered Runge--Kutta (RK) methods for nonlinear degenerate parabolic equations with space-, time-, state-, and gradient-dependent diffusion coefficients. The methods are linearly implicit in the sense that they evaluate nonlinear terms explicitly and require only linear solves. Unlike classical EIN methods that are direct applications of implicit--explicit (IMEX) RKs, our construction is built upon explicit RKs. The key ingredient is two-sided symmetric filtering applied at each RK stage: a high-order filter is applied both before and after evaluating the original nonlinear residual. It leads to a dissipation estimate independent of the time step and retains the underlying RK order formally. Assuming time-dependent splitting operators with stagewise pointwise coefficient bounds, we rigorously prove non-increasing $L^2$ norm for arbitrary time steps. A quadratic matrix criterion covers explicit RK-based methods with positive weights, while strong stability preserving (SSP) compositions yield exact energy identities and weighted dissipation estimates. We present SSP methods through fourth order and a method based on the classical RK4. The framework accommodates spatial discretizations with discrete gradient structures including continuous Galerkin, Bassi--Rebay 1, the method of Brezzi et al., and local discontinuous Galerkin. Compatible summation-by-parts discretizations with simultaneous approximation terms are also covered in their discrete mass norm.