A novel numerical method of high-order accuracy for flow in unsaturated porous media

作者：Zambra C E^*; Dumbser M; Toro E F; Moraga N O

来源：International Journal for Numerical Methods in Engineering, 2012, 89(2): 227-240.

DOI：10.1002/nme.3241

• unsaturated porous media
• soil watertransport
• novel numerical method
• ADER schemes
• space-time DGmethod

We construct finite volume schemes of very high order of accuracy in space and time for solving the nonlinear Richards equation (RE). The general scheme is based on a three-stage predictorcorrector procedure. First, a high-order weighted essentially non-oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level to guarantee monotonicity in the presence of steep gradients. Second, the temporal evolution of the WENO reconstruction polynomials is computed in a predictor stage by using a global weak form of the governing equations. A global spacetime DG FEM is used to obtain a scheme without the parabolic time-step restriction caused by the presence of the diffusion term in the RE. The resulting nonlinear algebraic system is solved by a NewtonKrylov method, where the generalized minimal residual method algorithm of Saad and Schulz is used to solve the linear subsystems. Finally, as a third step, the cell averages of the finite volume method are updated using a one-step scheme, on the basis of the solution calculated previously in the spacetime predictor stage. Our scheme is validated against analytical, experimental, and other numerical reference solutions in four test cases. A numerical convergence study performed allows us to show that the proposed novel scheme is high order accurate in space and time.

• 出版日期2012-1-13