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On the Stability of IMEX Upwind gSBP Schemes for 1D Linear Advection-Diffusion Equations
A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind...
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A Hybrid SBP-SAT/Fourier Pseudo-spectral Method for the Transient Wigner Equation Involving Inflow Boundary Conditions
In this paper, a hybrid SBP-SAT/pseudo-spectral method is proposed for solving the time-dependent Wigner equation. High-order summation-by-parts...
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Artificial Viscosity for SBP-FR Methods and the Construction of Fully Discrete Stable Schemes
In the last section of Chapter 5, we have seen that the artificial dissipation/spectral viscosity (or modal filters) can have an huge influence of...
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Numerical Schemes for Hyperbolic Problems
In this chapter, we present numerical methods for hyperbolic conservation laws and theoretical results for different types of schemes. There exists...
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Inverses of SBP-SAT Finite Difference Operators Approximating the First and Second Derivative
The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference...
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On an Eigenvalue Property of Summation-By-Parts Operators
Summation-By-Parts (SBP) methods provide a systematic way of constructing provably stable numerical schemes. However, many proofs of convergence and...
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A Unifying Algebraic Framework for Discontinuous Galerkin and Flux Reconstruction Methods Based on the Summation-by-Parts Property
We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for...
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Long-time Error Behavior of Discontinuous Galerkin and Flux Reconstruction
The investigation of the error behavior of numerical solutions to hyperbolic conservation laws is quite important and has received much interest in...
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Using the Dafermos entropy rate criterion in numerical schemes
The following work concerns the construction of an entropy dissipative finite volume solver based on the convex combination of an entropy...
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Multigrid Schemes for High Order Discretizations of Hyperbolic Problems
Total variation diminishing multigrid methods have been developed for first order accurate discretizations of hyperbolic conservation laws. This...
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New Stable Discontinuous Galerkin Methods on Equidistant and Scattered Points
Up to now, we have studied already existing numerical methods, investigated their properties and extended stability conditions in this context....
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Introduction
Many phenomena in nature and engineering science are described using partial differential equations (PDEs). Among all of them, hyperbolic...
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A Non-Stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate...
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Extension of Tensor-Product Generalized and Dense-Norm Summation-by-Parts Operators to Curvilinear Coordinates
Methodologies are presented that enable the construction of provably linearly stable and conservative high-order discretizations of partial...
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Optimal error estimates of an IPDG scheme for the resistive magnetic induction equation
In this paper, we develop the framework for error analysis of a fully-discrete interior penalty discontinuous Galerkin (IPDG) scheme designed for the...
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Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws
This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space–time...
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Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations
We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order...
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Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties
The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the...
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A New Discontinuous Galerkin Method for Elastic Waves with Physically Motivated Numerical Fluxes
The discontinuous Galerkin (DG) method is an established method for computing approximate solutions of partial differential equations in many...
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Entropy Stable Galerkin Methods with Suitable Quadrature Rules for Hyperbolic Systems with Random Inputs
In this paper, we investigate hyperbolic systems with random inputs based on generalized polynomial chaos (gPC) approximations, which is one of the...
