Journal of Approximation Theory
On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation
Journal of Computational Physics
Computational geometry in C
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
Journal of Computational Physics
Weighted essentially non-oscillatory schemes on triangular meshes
Journal of Computational Physics
Two Barriers on Strong-Stability-Preserving Time Discretization Methods
Journal of Scientific Computing
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
Fast high order ADER schemes for linear hyperbolic equations
Journal of Computational Physics
Finite-volume WENO schemes for three-dimensional conservation laws
Journal of Computational Physics
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
Derivative Riemann solvers for systems of conservation laws and ADER methods
Journal of Computational Physics
Journal of Computational Physics
ADER finite volume schemes for nonlinear reaction--diffusion equations
Applied Numerical Mathematics
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An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline weighted essentially non-oscillatory (WENO) reconstruction with high order flux evaluation. Polyharmonic splines are utilized in the recovery step of the finite volume method yielding a WENO reconstruction that is stable, flexible, and optimal in the associated Sobolev (Beppo-Levi) space. The flux evaluation is accomplished by solving generalized Riemann problems across cell interfaces. The mesh adaptation is performed through an a posteriori error indicator, which relies on the polyharmonic spline reconstruction scheme. The performance of the proposed method is illustrated by a series of numerical experiments, including linear advection, Burgers's equation, Smolarkiewicz's deformational flow test, and the five-spot problem.