Very efficient high-order hyperbolic schemes for time-dependent advection–diffusion problems: Third-, fourth-, and sixth-order

Abstract

In this paper, we construct very efficient high-order schemes for general time-dependent advection–diffusion problems, based on the first-order hyperbolic system method. Extending the previous work on the second-order time-dependent hyperbolic advection–diffusion scheme (Mazaheri and Nishikawa, NASA/TM-2014-218175, 2014), we construct third-, fourth-, and sixth-order accurate schemes by modifying the source term discretization. In this paper, two techniques for the source term discretization are proposed; (1) reformulation of the source terms with their divergence forms and (2) correction to the trapezoidal rule for the source term discretization. We construct spatially third- and fourth-order schemes from the former technique. These schemes require computations of the gradients and second-derivatives of the source terms. From the latter technique, we construct spatially third-, fourth-, and sixth-order schemes by using the gradients and second-derivatives for the source terms, except the fourth-order scheme, which does not require the second derivatives of the source term and thus is even less computationally expensive than the third-order schemes. We then construct high-order time-accurate schemes by incorporating a high-order backward difference formula as a source term. These schemes are very efficient in that high-order accuracy is achieved for both the solution and the gradient only by the improved source term discretization. A very rapid Newton-type convergence is achieved by a compact second-order Jacobian formulation. The numerical results are presented for both steady and time-dependent linear and nonlinear advection–diffusion problems, demonstrating these powerful features.