Insights from von Neumann analysis of high-order flux reconstruction schemes

2011 ◽  
Vol 230 (22) ◽  
pp. 8134-8154 ◽  
Author(s):  
P.E. Vincent ◽  
P. Castonguay ◽  
A. Jameson
Keyword(s):  
High Order ◽  
Flux Reconstruction ◽  
Von Neumann ◽  
Von Neumann Analysis

scholarly journals High-order compact LOD methods for solving high-dimensional advection equations

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Bo Hou ◽  
Yongbin Ge
Keyword(s):  
Truncation Error ◽  
Three Dimensional ◽  
Taylor Series Expansion ◽  
High Order ◽  
High Dimensional ◽  
Analysis Method ◽  
Order Accuracy ◽  
Von Neumann ◽  
One Dimensional ◽  
Von Neumann Analysis

AbstractIn this paper, by using the local one-dimensional (LOD) method, Taylor series expansion and correction for the third derivatives in the truncation error remainder, two high-order compact LOD schemes are established for solving the two- and three- dimensional advection equations, respectively. They have the fourth-order accuracy in both time and space. By the von Neumann analysis method, it shows that the two schemes are unconditionally stable. Besides, the consistency and convergence of them are also proved. Finally, numerical experiments are given to confirm the accuracy and efficiency of the present schemes.

scholarly journals Análisis de Von Neumann para el métodoLocal Discontinuous Galerkin en 1D

2019 ◽  
Vol 37 (2) ◽  
pp. 199-217
Author(s):  
Paul Castillo ◽  
Sergio Gómez
Keyword(s):  
Discontinuous Galerkin ◽  
Numerical Experiments ◽  
High Order ◽  
Stability Conditions ◽  
Von Neumann ◽  
Theoretical Tool ◽  
Von Neumann Analysis ◽  
Time Marching ◽  
The Stability ◽  
Theoretical Results

Using the von Neumann analysis as a theoretical tool, an analysisof the stability conditions of some explicit time marching schemes, in com-bination with the spatial discretizationLocal Discontinuous Galerkin(LDG)and high order approximations, is presented. The stabilityconstant, CFL(Courant-Friedrichs-Lewy), is studied as a function of theLDG parametersand the approximation degree. A series of numerical experiments is carriedout to validate the theoretical results.

A High-Order Flux Reconstruction Method for 2-D Vorticity Transport

2021 ◽  
Author(s):  
Adrin Gharakhani
Keyword(s):  
Finite Difference ◽  
Poisson Equation ◽  
Finite Difference Methods ◽  
Neumann Boundary ◽  
Normal Derivative ◽  
High Order ◽  
Benchmark Problems ◽  
Flux Reconstruction ◽  
Reconstruction Method ◽  
Vorticity Transport

Abstract A compact high-order finite difference method on unstructured meshes is developed for discretization of the unsteady vorticity transport equations (VTE) for 2-D incompressible flow. The algorithm is based on the Flux Reconstruction Method of Huynh [1, 2], extended to evaluate a Poisson equation for the streamfunction to enforce the kinematic relationship between the velocity and vorticity fields while satisfying the continuity equation. Unlike other finite difference methods for the VTE, where the wall vorticity is approximated by finite differencing the second wall-normal derivative of the streamfunction, the new method applies a Neumann boundary condition for the diffusion of vorticity such that it cancels the slip velocity resulting from the solution of the Poisson equation for the streamfunction. This yields a wall vorticity with order of accuracy consistent with that of the overall solution. In this paper, the high-order VTE solver is formulated and results presented to demonstrate the accuracy and convergence rate of the Poisson solution, as well as the VTE solver using benchmark problems of 2-D flow in lid-driven cavity and backward facing step channel at various Reynolds numbers.

scholarly journals High-Order Flux Reconstruction on Stretched and Warped Meshes

AIAA Journal ◽  
2019 ◽  
Vol 57 (1) ◽  
pp. 341-351 ◽  
Author(s):  
Will Trojak ◽  
Rob Watson ◽  
Paul G. Tucker
Keyword(s):  
High Order ◽  
Flux Reconstruction

scholarly journals On the exact and numerical solutions to a nonlinear model arising in mathematical biology

2018 ◽  
Vol 22 ◽  
pp. 01061 ◽  
Author(s):  
Asif Yokus ◽  
Tukur Abdulkadir Sulaiman ◽  
Haci Mehmet Baskonus ◽  
Sibel Pasali Atmaca
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Hyperbolic Function ◽  
Von Neumann ◽  
Convection Diffusion Equation ◽  
Wolfram Mathematica ◽  
Forward Difference ◽  
Von Neumann Analysis ◽  
The Stability

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ(η))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L2 and L∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

Connections between the discontinuous Galerkin method and high-order flux reconstruction schemes

2014 ◽  
Vol 75 (12) ◽  
pp. 860-877 ◽  
Author(s):  
D. De Grazia ◽  
G. Mengaldo ◽  
D. Moxey ◽  
P. E. Vincent ◽  
S. J. Sherwin
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
High Order ◽  
Flux Reconstruction

A New Class of High-Order Energy Stable Flux Reconstruction Schemes

2010 ◽  
Vol 47 (1) ◽  
pp. 50-72 ◽  
Author(s):  
P. E. Vincent ◽  
P. Castonguay ◽  
A. Jameson
Keyword(s):  
High Order ◽  
Flux Reconstruction ◽  
New Class ◽  
Order Energy ◽  
Energy Stable

Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM35-SM46 ◽  
Author(s):  
Matthew M. Haney
Keyword(s):  
Wave Propagation ◽  
Numerical Model ◽  
Finite Difference ◽  
Stability Criterion ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
Von Neumann ◽  
Transmission Coefficients ◽  
Strong Material ◽  
Von Neumann Analysis

Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model — that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable.

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