Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws.

*(English)*Zbl 1142.65070Summary: We propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First a high-order weighted essentially non-oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed locally inside each cell using the governing equations. In the original ENO scheme of A. Harten, B. Engquist, S.Osher, and S. Chakravarthy [ibid. 71, 231–303 (1987; Zbl 0652.65067)] and in the ADER schemes of V. A. Titarev and E. F. Toro [ibid. 204, No. 2, 715–736 (2005; Zbl 1060.65641)], this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing partial differential equation with respect to space and time, i.e. by applying the so-called Cauchy-Kovalewski procedure.

However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy-Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space-time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space-time integration over each control volume, using the local space-time DG solutions at the Gaussian integration points for the intercell fluxes and for the space-time integral over the source term.

We show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of R. J. LeVeque and H. C. Yee [ibid. 86, No. 1, 187–210 (1990; Zbl 0682.76053)], the relaxation system of S. Jin and Z. Xin, [Commun. Pure Appl. Math. 48, No. 3, 235–276 (1995; Zbl 0826.65078)] and the full compressible Euler equations with stiff friction source terms.

However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy-Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space-time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space-time integration over each control volume, using the local space-time DG solutions at the Gaussian integration points for the intercell fluxes and for the space-time integral over the source term.

We show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of R. J. LeVeque and H. C. Yee [ibid. 86, No. 1, 187–210 (1990; Zbl 0682.76053)], the relaxation system of S. Jin and Z. Xin, [Commun. Pure Appl. Math. 48, No. 3, 235–276 (1995; Zbl 0826.65078)] and the full compressible Euler equations with stiff friction source terms.

##### MSC:

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

35L65 | Hyperbolic conservation laws |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

hyperbolic balance laws; stiff source terms; finite volume schemes; ADER approach; local space-time discontinuous Galerkin method; WENO reconstruction; weighted essentially non-oscillatory (WENO); algorithm; finite element; compressible Euler equations; numerical examples
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\textit{M. Dumbser} et al., J. Comput. Phys. 227, No. 8, 3971--4001 (2008; Zbl 1142.65070)

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