FINITE-VOLUME IMPLEMENTATION OF HIGH-ORDER ESSENTIALLY NONOSCILLATORY SCHEMES IN 2 DIMENSIONS

We continue the study of the finite-volume application of high-order-accurate, essentially nonoscillatory shock-capturing schemes to two-dimensional initial-boundary-value problems. These schemes achieve high-order spatial accuracy, in smooth regions, by a piecewise polynomial approximation of the solution from cell averages. In addition, this spatial operation involves an adaptive stencil algorithm, in order to avoid the oscillatory behavior that is associated with interpolation across steep gradients. High-order Runge-Kutta methods are employed for time integration, thus making these schemes best suited for unsteady problems. Schemes are developed which use fifth- and sixth-order-accurate spatial operators in conjunction with a fourth-order time operator. Under a sufficient time-step restriction, numerical results suggest that these schemes converge according to the higher-order spatial accuracy, for unsteady problems. Second-, third-, and fifth-order algorithms are applied to the Euler equations of gasdynamics and tested on a problem that models a shock-vortex interaction.