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

*(English)*Zbl 1390.65078Summary: 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 our previous work on the second-order time-dependent hyperbolic advection-diffusion scheme [First-order hyperbolic system method for time-dependent advection-diffusion problems. Defense Technical Information Center (2014; doi:10.21236/ada606728)], 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.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Software:

IHEAT
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\textit{A. Mazaheri} and \textit{H. Nishikawa}, Comput. Fluids 102, 131--147 (2014; Zbl 1390.65078)

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##### References:

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