Dispersion Analysis for Discontinuous Spectral Element Methods
Journal of Scientific Computing
Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations
Journal of Computational Physics
Nodal high-order methods on unstructured grids
Journal of Computational Physics
Polymorphic nodal elements and their application in discontinuous Galerkin methods
Journal of Computational Physics
Stable and Accurate Interpolation Operators for High-Order Multiblock Finite Difference Methods
SIAM Journal on Scientific Computing
An error minimized pseudospectral penalty direct Poisson solver
Journal of Computational Physics
High-fidelity numerical solution of the time-dependent Dirac equation
Journal of Computational Physics
Optimal diagonal-norm SBP operators
Journal of Computational Physics
A generalized framework for nodal first derivative summation-by-parts operators
Journal of Computational Physics
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This paper, concluding the trilogy, develops schemes for the stable solution of wave-dominated unsteady problems in general three-dimensional domains. The schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established. The complex computational domains are constructed by using nonoverlapping quadrilaterals in the two-dimensional case and hexahedrals in the three-dimensional space. To illustrate the ideas underlying the multidomain method, a stable scheme for the solution of the three-dimensional linear advection-diffusion equation in general curvilinear coordinates is developed. The analysis suggests a novel, yet simple, stable treatment of geometric singularities like edges and vertices. The theoretical results are supported by a two-dimensional implementation of the scheme.The main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible Navier--Stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented. Well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume, with special attention given to a hexahedral domain. The treatment includes a stable procedure for dealing with boundary conditions at a solid wall.The efficacy of the scheme for the compressible Navier--Stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions. The results are found to agree with the solution of the compressible boundary layer equations.