Finite difference method

Abstract: The point-source traveltime field has an upwind singularity at the source point. Consequently, all formally high-order, finite-difference eikonal solvers exhibit first-order convergence and relatively large errors. Adaptive upwind finite-difference methods based on high-order Weighted Essentially NonOscillatory (WENO) Runge-Kutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on a posteriori error estimation. It achieves prescribed accuracy at a far lower cost than does the fixed-grid method. Moreover, the achieved high accuracy of traveltimes yields reliable estimates of auxiliary quantities such as take-off angles and geometric spreading factors.

Abstract: The paper discusses the use of the artificial compression method for the computation of discontinuous solutions of a single conservation law by finite difference methods. The single conservation law has either a shock or a contact discontinuity. Any monotone finite difference scheme applied to the original equation smears the discontinuity, while the same scheme applied to the equation modified by an artificial compression flux produces steady progressing profiles. If L is any finite difference scheme in conservation form and C is an artificial compressor, the split flux artificial compression method CL is a corrective scheme: L smears the discontinuity while propagating it; C compresses the smeared transition toward a sharp discontinuity. Numerical implementation of artificial compression is described.

Abstract: SUMMARY An efficient numerical scheme is outlined for solving the SWEs (shallow water equations) in environmental flow; this scheme includes the addition of a five-point symmetric total variation diminishing (TVD) term to the corrector step of the standard MacCormack scheme. The paper shows that the discretization of the conservative and non-conservative forms of the SWEs leads to the same finite difference scheme when the source term is discretized in a certain way. The non-conservative form is used in the solution outlined herein, since this formulation is simpler and more efficient. The time step is determined adaptively, based on the maximum instantaneous Courant number across the domain. The bed friction is included either explicitly or implicitly in the computational algorithm according to the local water depth. The wetting and drying process is simulated in a manner which complements the use of operator-splitting and two-stage numerical schemes. The numerical model was then applied to a hypothetical dam-break scenario, an experimental dam-break case and an extreme flooding event over the Toce River valley physical model. The predicted results are free of spurious oscillations for both sub- and super-critical flows, and the predictions compare favourably with the experimental measurements. Copyright q 2006 John Wiley & Sons, Ltd.