Solution of the generalized Riemann problem for advection–reaction equations

By Eleuterio F. Toro, Vladimir A. Titarev. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2002

257 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/466e9a541f2f6036b9e2e917a1f04b2575cbd7a0

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In Applied Mathematics

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Citations in the corpus

A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes (2008, methodology): “For nonlinear systems with source terms a semi-analytical solver was first proposed by Toro and Titarev, see [73,70,71].”
An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes — II. The three-dimensional isotropic case (2006, methodology): “The main ingredients of the ADER approach are a Taylor expansion in time, the solution of Derivative Riemann Problems (DRP) (Toro & Titarev 2002) to approximate the space derivatives at the interface and the Cauchy-Kovalewski procedure for replacing the time derivatives in the Taylor series by…”
Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems (2007, methodology): “on ENO schemes [27] and furthermore in the family of ADER-FV and ADER-DG schemes [17,21,33,47,49,51], the use of the Cauchy–Kovalewski procedure to avoid numerical quadrature in space and time is new.”
ADER: Arbitrary High Order Godunov Approach (2002, methodology): “Next, it can be shown [11, 13 ] that space derivatives Q (k) — “ (k) Q of the solution at (x − xi+1/2)/y=0 can be evaluated as the Godunov states of the the following linearised GRP0:”
The discontinuous Galerkin method with Lax-Wendroff type time discretizations (2005, methodology): “Dumbser and Munz [14] are also extending the ADER discontinuous Galerkin method to the nonlinear case using generalized Riemann solvers [36].”
Godunov-type methods for free-surface shallow flows: A review (2007, methodology): “This Riemann problem has been termed the Derivative Riemann Problem, or DRP (Toro and Titarev, 2006), see also Toro and Titarev (2002).”
A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws (2014, methodology): “The above iterative procedure has replaced the cumbersome Cauchy-Kovalewski procedure that has been initially employed in the original version of the ADER finite volume and ADER discontinuous Galerkin schemes [93, 102, 105, 103, 39, 101, 38].”
Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems (2007, methodology): “This is described in more detail in [11,15,44,41,45] for the nonlinear case.”
ADER schemes for three-dimensional non-linear hyperbolic systems (2005, methodology): “To evaluate the numerical flux, in the ADER approach one solves the generalised Riemann problem with initial condition consisting of two arbitrary but smooth functions using a semi-analytical method, reported in [ 32 ].”
Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed (2009, methodology): “In order to obtain a high order accurate one-step method we use the ADER approach of Titarev and Toro [30].”