Second-order accurate monotone finite volume scheme for Richards’ equation

In this work we perform a theoretical and numerical analysis of Richards’ equation. For certain types of nonlinearities we provide explicit analytical solutions. These solutions are used to show that conventional unconditionally monotone finite volume schemes have only first-order accuracy. We derive necessary and sufficient conditions for the monotonicity of finite volume discretizations and use these conditions to construct a monotone finite volume discretization accurate to second-order.