High-Order Numerical Methods for Wave Equations with van der Pol Type Boundary Conditions

We develop high-order numerical methods for solving wave equations with van der Pol type nonlinear boundary conditions. Based on the wave reflection on the boundaries, we first solve the corresponding Riemann invariants by constructing two iterative mappings, and then, regarding the regularity of boundary conditions, propose two different high-order numerical approaches to the system. When the degree of regularity is high, we establish a sixth-order finite difference scheme. While for a low degree of regularity, we provide another method by utilizing the high-order GaussKronrod quadrature rule. Numerical experiments are performed to illustrate the proposed approaches.