Study Of Implicit Runge-kutta Algorithm For Richards Equation

Due to the increasing demand for water resources development and utilization in today’s society,the development of powerful,accurate and efficient solvers are essential to predict unsaturated soil water movement.The Richards equation for unsaturated soil water movement is a highly nonlinear parabolic partial differential equation,which is difficult to solve directly.After decades of research,numerical simulation methods have been widely used to solve the Richards equation,which is now the main method to study unsaturated soil water movement.Therefore,a more efficient and accurate numerical solution of the Richards equation is used to simulate the unsaturated soil water movement.The finite volume method is used to discretize the spatial terms of the Richards equation.In addition to using the fully implicit first order backward difference formula and second order backward difference formula for temporal integration,an implicit Runge-Kutta algorithm with adaptive time stepping strategy is introduced for temporal integration.In order to simulate the solution of the Richards equation,the Newton–Krylov method is introduced to solve the nonlinear system generated at each time step.The linear system corresponding to each Newton iteration step is usually large sparse and ill-conditioned,thus leading to slow convergence or even non-convergence of the fully implicit algorithm.This problem can be effectively solved by using reasonable preconditioners for linear system.The restricted additive Schwarz preconditioner is derived from the principle of domain decomposition.The domain decomposition method is a technique based on the “divide and conquer” principle,which divides the computational domain into subdomains and solves the problem by solving subproblem on each subdomain,thus reducing the difficulty of solving the problem.The restricted additive Schwarz preconditioner applied to the linear system accelerate the convergence of linear iterations,saves the communication costs and improves the robustness of Newton method.Finally,several numerical results related to the test cases are given to illustrate the efficiency,robustness of the Newton–Krylov algorithm.In the given test cases,the performance of fully implicit Runge-Kutta algorithm combined with fixed time stepping strategy and adaptive time stepping strategy is compared.The numerical accuracy of the fully implicit Runge-Kutta algorithm with respect to the time convergence order is further investigated.Under the condition of using the second-order implicit Runge-Kutta algorithm,the computation time graph and the nonlinear iteration number graph are drawn.Numerical results implicate that the implicit Runge-Kutta algorithm used in the test cases significantly enhance the performance of Newton–Krylov–Schwarz(NKS)algorithm.