ADER: Arbitrary High Order Godunov Approach

By Vladimir A. Titarev, Eleuterio F. Toro. Journal of Scientific Computing, 2002

458 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/cf3bc2ee6edbf895fb72dfa714ee2e0270944d08

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Citations in the corpus

Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems (2007, methodology): “The extension to nonlinear hyperbolic conservation laws with source terms has then been achieved by Titarev and Toro using a WENO reconstruction technique on Cartesian grids [40,44,41,45].”
On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws (2011, methodology): “[24], in the ADER schemes of Titarev and Toro [40, 41] and in the ADER-DG & LW-DG approaches presented in [15, 33].”
The discontinuous Galerkin method with Lax-Wendroff type time discretizations (2005, methodology): “More recently, Dumbser [13] developed the ADER (Arbitrary high order schemes using DERivatives, see [34]) discontinuous Galerkin method for solving the linear aeroacoustics systems.”
High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations (2004, methodology): “However, as pointed out by V.A. Titarev and E.F. Toro [26], the design of high order accurate numerical schemes for hyperbolic conservation laws is a formidable task since three major difficulties have to be overcome: ensuring the conservation property, preserving the high order of accuracy in both ”
An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes — II. The three-dimensional isotropic case (2006, methodology): “…numerical method relies on the combination of the ADER approach, originally developed by Toro et al. (Toro 2001; Schwartzkopff, Munz & Toro 2004; Titarev & Toro 2002; Schwartzkopff, Dumbser & Munz 2002) in the Þnite volume (FV) framework, and the Discontinuous Galerkin Þnite element method.”
Godunov-type methods for free-surface shallow flows: A review (2007, background): “See also Dumbser (2005), Dumbser and Munz (2005), Käser (2003, 2004), Käser and Iske (2005), Takakura and Toro (2002), Titarev and Toro (2002, 2005) and Toro and Titarev (2005).”
ADER schemes for three-dimensional non-linear hyperbolic systems (2005, methodology): “The first option is the state-expansion ADER [26], in which the approximate state (5) is inserted into the definition of the numerical flux (3) and then an appropriate r-order accurate quadrature is used for time integration: Fi+1/2 = Kl ∑”
An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes — III. Viscoelastic attenuation (2007, methodology): “Originally developed in (Reed & Hill 1973), Dumbser (2005) and Dumbser & Munz (2005a; 2005b) combined the DG method with the ADER approach of (Toro, Millington & Nejad 2001; Titarev & Toro 2002; Toro & Titarev 2002) for accurate time integration using Arbitrary high order DERivatives.”
Solution of the generalized Riemann problem for advection–reaction equations (2002, methodology): “…be used to construct numerical methods of very high order of accuracy in space and time to solve the general initial-boundary-value problem; results reported in Toro et al. (2001), Toro & Titarev (2001a,b), Titarev (2001), Titarev & Toro (2002) and Schwartzkopff et al. (2001) look very promising.”
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].”
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes (2008, methodology): “[72,69,73,63,70,71,49,32,67,28,29] and other CauchyKovalewski procedure based methods [34,57,56,52,37].”