The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
On the numerical dissipation of high resolution schemes for hyperbolic conservation laws
Journal of Computational Physics
Spectral methods on triangles and other domains
Journal of Scientific Computing
A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems
Journal of Computational Physics
On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation
Journal of Computational Physics
Local mesh adaptation in two space dimensions
IMPACT of Computing in Science and Engineering
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
High resolution schemes for hyperbolic conservation laws
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
Journal of Computational Physics
Weighted essentially non-oscillatory schemes on triangular meshes
Journal of Computational Physics
Journal of Computational Physics
A technique of treating negative weights in WENO schemes
Journal of Computational Physics
A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation
Journal of Computational Physics
ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D
Journal of Scientific Computing
ADER: Arbitrary High Order Godunov Approach
Journal of Scientific Computing
Journal of Computational Physics
Fast high order ADER schemes for linear hyperbolic equations
Journal of Computational Physics
Runge--Kutta Discontinuous Galerkin Method Using WENO Limiters
SIAM Journal on Scientific Computing
Journal of Computational Physics
ADER schemes for three-dimensional non-linear hyperbolic systems
Journal of Computational Physics
ADER schemes on adaptive triangular meshes for scalar conservation laws
Journal of Computational Physics
TVD Fluxes for the High-Order ADER Schemes
Journal of Scientific Computing
Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes
Journal of Scientific Computing
Journal of Computational Physics
Solvers for the high-order Riemann problem for hyperbolic balance laws
Journal of Computational Physics
A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes
Journal of Computational Physics
Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws
Journal of Computational Physics
Runge-Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
FORCE schemes on unstructured meshes I: Conservative hyperbolic systems
Journal of Computational Physics
Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics
Journal of Computational Physics
Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations
Journal of Computational Physics
Nonuniform time-step Runge-Kutta discontinuous Galerkin method for Computational Aeroacoustics
Journal of Computational Physics
WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions
Journal of Computational Physics
Approximation error of the Lagrange reconstructing polynomial
Journal of Approximation Theory
Journal of Scientific Computing
Implicit high-order method for calculating rarefied gas flow in a planar microchannel
Journal of Computational Physics
Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction
SIAM Journal on Scientific Computing
A class of hybrid DG/FV methods for conservation laws II: Two-dimensional cases
Journal of Computational Physics
Visualization of Advection-Diffusion in Unsteady Fluid Flow
Computer Graphics Forum
Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms
Journal of Computational Physics
Modeling and numerical approximation of a 2.5D set of equations for mesoscale atmospheric processes
Journal of Computational Physics
Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics
Journal of Computational Physics
A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods
Journal of Computational Physics
A Robust Reconstruction for Unstructured WENO Schemes
Journal of Scientific Computing
WENO schemes on arbitrary unstructured meshes for laminar, transitional and turbulent flows
Journal of Computational Physics
High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows
Journal of Computational Physics
A gradient augmented level set method for unstructured grids
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.57 |
In this article we present a non-oscillatory finite volume scheme of arbitrary accuracy in space and time for solving linear hyperbolic systems on unstructured grids in two and three space dimensions using the ADER approach. The key point is a new reconstruction operator that makes use of techniques developed originally in the discontinuous Galerkin finite element framework. First, we use a hierarchical orthogonal basis to perform reconstruction. Second, reconstruction is not done in physical coordinates, but in a reference coordinate system which eliminates scaling effects and thus avoids ill-conditioned reconstruction matrices. In order to achieve non-oscillatory properties, we propose a new WENO reconstruction technique that does not reconstruct point-values but entire polynomials which can easily be evaluated and differentiated at any point. We show that due to the special reconstruction the WENO oscillation indicator can be computed in a mesh-independent manner by a simple quadratic functional. Our WENO scheme does not suffer from the problem of negative weights as previously described in the literature, since the linear weights are not used to increase accuracy. Accuracy is obtained by merely putting a large linear weight on the central stencil. The resulting one-step ADER finite volume scheme obtained in this way performs only one nonlinear WENO reconstruction per element and time step and thus can be implemented very efficiently even for unstructured grids in three space dimensions. We show convergence results obtained with the proposed method up to sixth order in space and time on unstructured triangular and tetrahedral grids in two and three space dimensions, respectively.