Journal of Computational Physics
Spectral methods on triangles and other domains
Journal of Scientific Computing
A conservative staggered-grid Chebyshev multidomain method for compressible flows
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Spectral difference method for unstructured grids I: basic formulation
Journal of Computational Physics
Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations
Journal of Scientific Computing
On the Stability and Accuracy of the Spectral Difference Method
Journal of Scientific Computing
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
A Study of Viscous Flux Formulations for a p-Multigrid Spectral Volume Navier Stokes Solver
Journal of Scientific Computing
A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy
Journal of Scientific Computing
LDG2: A Variant of the LDG Flux Formulation for the Spectral Volume Method
Journal of Scientific Computing
A New Class of High-Order Energy Stable Flux Reconstruction Schemes
Journal of Scientific Computing
Journal of Computational Physics
A p-adaptive LCP formulation for the compressible Navier-Stokes equations
Journal of Computational Physics
Journal of Scientific Computing
Energy stable flux reconstruction schemes for advection-diffusion problems on triangles
Journal of Computational Physics
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The flux reconstruction (FR) approach allows various well-known high-order schemes, such as collocation based nodal discontinuous Galerkin (DG) methods and spectral difference (SD) methods, to be cast within a single unifying framework. Recently, the authors identified a new class of FR schemes for 1D conservation laws, which are simple to implement, efficient and guaranteed to be linearly stable for all orders of accuracy. The new schemes can easily be extended to quadrilateral elements via the construction of tensor product bases. However, for triangular elements, such a construction is not possible. Since numerical simulations over complicated geometries often require the computational domain to be tessellated with simplex elements, the development of stable FR schemes on simplex elements is highly desirable. In this article, a new class of energy stable FR schemes for triangular elements is developed. The schemes are parameterized by a single scalar quantity, which can be adjusted to provide an infinite range of linearly stable high-order methods on triangular elements. Von Neumann stability analysis is conducted on the new class of schemes, which allows identification of schemes with increased explicit time-step limits compared to the collocation based nodal DG method. Numerical experiments are performed to confirm that the new schemes yield the optimal order of accuracy for linear advection on triangular grids.