Comparison of discretization accuracy in an adaptive multigrid barotropic tropical cyclone track model

Among the many sources of error in modeling tropical cyclones is discretization error: how well are the governing equations approximated numerically? While discretization error is typically small, numerical methods with smaller errors should be preferred over others, all else being equal. This paper addresses the tradeoff between higher-accuracy discretization and increased computational work in an adaptive multigrid tropical cyclone track model (MUDBAR). The model achieves higher resolution near the vortex by superimposing nested overlapping grids with different mesh sizes. Unlike conventional nested-grid methods, multigrid processing allows optimal solution speed and accurate estimates of truncation error; the latter are used in an adaptive mesh refinement scheme to provide just the resolution needed at each point.

Previous versions of the model used second-order finite differences; now we investigate the gains possible by using fourth-order differencing. In particular, we: (1) compare fourth-order multigrid solvers for the Helmholtz problem relating vorticity and streamfunction, (2) show that no compact fourth-order generalization of the Arakawa Jacobian exists, and (3) compare fourth-order noncompact Jacobians and extrapolation techniques for the advection terms in the barotropic vorticity equation. Numerical experiments are used to evaluate the improvements in accuracy and efficiency over second-order differencing.