HIGH-ORDER FINITE VOLUME APPROXIMATIONS OF DIFFERENTIAL-OPERATORS ON NONUNIFORM GRIDS

When computing numerical solutions to partial differential equations, difference operators that mimic the crucial properties of the differential operators are usually more accurate than those that do not. Properties such as symmetry, conservation, stability, and the duality relationships and identities between the gradient, curl, and divergence operators are all important. Using the finite volume method, we have derived local, accurate, reliable and efficient difference methods that mimic these properties on nonuniform rectangular and cuboid grids. In a finite volume method, the divergence, gradient, and curl operators are defined using a discrete versions of the divergence theorem and Stokes' theorem. These methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks. Numerical examples comparing local second and fourth-order finite volume approximations to conservation laws on very rough grids are used to demonstrate the advantages of the higher order methods.