Next: EIKONAL EQUATION Up: VISCOSITY SOLUTIONS AND UPWIND Previous: Conservation laws

## Upwind finite differences

Conservation laws are commonly solved by upwind finite differencing, a numerical technique that plays an important role in modern computational fluid dynamics; see for example Harten et al. (1987) for a review. Depending on the flow direction, upwind finite-difference schemes use a backward or forward finite-difference approximation to the derivative operator. The sense of the derivative is switched so that only field points from upstream in the flow are used to update any given point.

This direction switching is forced by the appearance of discontinuities (shocks) in the flow variables. For the eikonal equation, the gradient components of traveltime can be discontinuous: rays, whose direction is determined by the gradient components, ``break'' at interfaces in the model where velocity contrasts occur. Reshef and Kosloff (1986) observe stability problems in a centered finite-difference scheme at these interfaces. Upwind finite-difference methods, on the other hand, are stable because they mimic the underlying physics of the problem in two important aspects. First, they add numerical viscosity to the equations, thus finding a smooth viscosity solution. Second, they copy the behavior of continuum flow by taking their information from upstream.

Careful inspection of Vidale's ordering of gridpoints reveals that he applies the basic ``box'' (four-point) difference formula only in the upwind sense (see Vidale, 1988, Figure 5), having used an alternative extrapolation at the few points at which the flow (i.e. ray field) changes direction along the computational front. Those points are local traveltime minima. His scheme is thus an upwind finite-difference method, although not presented as such. It is an implicit method, as it connects more than one value on the grid level being updated. The ordering of points is required to achieve a closed-form solution of the difference formulas, as opposed to an iterative approximate solution (as is often chosen with implicit schemes). Our formulas are explicit, in contrast, and do not require any special ordering of points.

Next: EIKONAL EQUATION Up: VISCOSITY SOLUTIONS AND UPWIND Previous: Conservation laws
Stanford Exploration Project
1/13/1998