Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
On Godunov-type methods for gas dynamics
SIAM Journal on Numerical Analysis
Efficient implementation of essentially non-oscillatory shock-capturing schemes,II
Journal of Computational Physics
A study of numerical methods for hyperbolic conservation laws with stiff source terms
Journal of Computational Physics
On Godunov-type methods near low densities
Journal of Computational Physics
SIAM Journal on Scientific Computing
Numerical methods for hyperbolic conservation laws with stiff relaxation I: spurious solutions
SIAM Journal on Applied Mathematics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
One-dimensional front tracking based on high resolution wave propagation methods
SIAM Journal on Scientific Computing
Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Numerical schemes for hyperbolic conservation laws with stiff relaxation terms
Journal of Computational Physics
A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
SIAM Journal on Numerical Analysis
Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation
SIAM Journal on Numerical Analysis
Approximate Riemann solvers, parameter vectors, and difference schemes
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method
SIAM Journal on Numerical Analysis
Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations
SIAM Journal on Scientific Computing
Numerical Schemes for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation
SIAM Journal on Numerical Analysis
Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations
Recent trends in numerical analysis
Numerical Recipes: FORTRAN
A Modified Fractional Step Method for the Accurate Approximation of Detonation Waves
SIAM Journal on Scientific Computing
ADER: Arbitrary High Order Godunov Approach
Journal of Scientific Computing
Equilibrium schemes for scalar conservation laws with stiff sources
Mathematics of Computation
Applied Numerical Mathematics
Finite Difference WENO Schemes with Lax--Wendroff-Type Time Discretizations
SIAM Journal on Scientific Computing
Well balanced finite volume methods for nearly hydrostatic flows
Journal of Computational Physics
Fast high order ADER schemes for linear hyperbolic equations
Journal of Computational Physics
Journal of Computational Physics
ADER schemes for three-dimensional non-linear hyperbolic systems
Journal of Computational Physics
Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes
Journal of Scientific Computing
Asymptotic preserving and positive schemes for radiation hydrodynamics
Journal of Computational Physics
Journal of Computational Physics
Upwinding of the source term at interfaces for Euler equations with high friction
Computers & Mathematics with Applications
A modified higher order Godunov's scheme for stiff source conservative hydrodynamics
Journal of Computational Physics
Journal of Computational Physics
ADER finite volume schemes for nonlinear reaction--diffusion equations
Applied Numerical Mathematics
Journal of Computational Physics
Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations
Journal of Computational Physics
A hybrid Godunov method for radiation hydrodynamics
Journal of Computational Physics
ADER Schemes for Nonlinear Systems of Stiff Advection---Diffusion---Reaction Equations
Journal of Scientific Computing
Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms
Journal of Computational Physics
Journal of Computational Physics
A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme
Journal of Computational Physics
A two-dimensional fourth-order unstructured-meshed Euler solver based on the CESE method
Journal of Computational Physics
The equilibrium state method for hyperbolic conservation laws with stiff reaction terms
Journal of Computational Physics
Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes
Journal of Computational Physics
Hi-index | 31.51 |
In this article, 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 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 Harten et al. and in the ADER schemes of Titarev and Toro, this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing PDE 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 will 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 LeVeque and Yee [R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of Computational Physics 86 (1) (1990) 187-210], the relaxation system of Jin and Xin [S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics 48 (1995) 235-277] and the full compressible Euler equations with stiff friction source terms.