A Second-Order Bound-Preserving Exponential Scheme for Degenerate Parabolic Equations
Chen Chang, Chi‐Wang Shu · SIAM Journal on Scientific Computing · 2025
Abstract. This paper proposes a second-order numerical method for solving nonlinear parabolic equations with degenerate mobility. The intrinsic degenerate mobility in the equation yields a globally bounded solution. A pivotal feature of our methodology is an appropriate reformulation of the equation into an equivalent form. After applying a discontinuous Galerkin spatial discretization method, we derive a fully nonlinear ordinary differential equation (ODE) with a splitting structure. By introducing a linear term into the ODE, an exponential temporal discretization method, which involves only linear solvers, is proposed based on integrating factors and strong stability preserving Runge–Kutta methods. Our approach is proven to exhibit second-order accuracy, ensures bound preservation and mass conservation, and demonstrates a favorable CFL condition [Formula: see text], where [Formula: see text] and [Formula: see text] are the temporal and spatial mesh sizes, respectively. Comprehensive numerical tests validate the second-order accuracy and bound-preserving behaviors of our method.