Bound-preserving schemes for \(P^2\) local discontinuous Galerkin discretizations of KdV-type equations

In this paper, we construct a bound-preserving (BP) local discontinuous Galerkin (LDG) method for the generalized third-order Korteweg–de Vries (KdV) equations. The KdV equations have been widely used in mathematical models in dispersive hydrodynamics. We are interested in a new algorithm that is closer to the real physical bounds of solitary waves and dispersive shock waves (DSWs). We design a BP scheme for \(P^2\) $P^2$ LDG discretizations on nonuniform meshes so that an appropriate choice for the time step, the space size, and the penalty term is allowed to make the cell average of numerical solutions within the scope of global bounds. The third-order strong stability preserving (SSP) Runge–Kutta method is used to preserve the BP property of the discrete scheme. Furthermore, we extend the BP scheme to the two-dimensional Zakharov–Kuznetsov (ZK) equation. Numerical experiments demonstrate a good performance and accuracy of the BP LDG method.