High order regularity for solutions of the inviscid burgers equation

1989 ◽  
pp. 147-154 ◽  
Author(s):  
Ronald A. DeVore ◽  
Bradley J. Lucier
Keyword(s):  
Burgers Equation ◽  
High Order ◽  
Inviscid Burgers Equation

On the Lie point symmetry analysis and solutions of the inviscid Burgers equation

Pramana ◽  
2011 ◽  
Vol 77 (3) ◽  
pp. 407-414 ◽  
Author(s):  
MUHAMMAD A ABDULWAHHAB ◽  
ASHFAQUE H BOKHARI ◽  
A H KARA ◽  
F D ZAMAN
Keyword(s):  
Burgers Equation ◽  
Symmetry Analysis ◽  
Point Symmetry ◽  
Inviscid Burgers Equation ◽  
Lie Point Symmetry

A FDM for Burgers’ Equation on Unbounded Domains Using High-Order Artificial Boundary Conditions

2017 ◽  
Vol 865 ◽  
pp. 233-238
Author(s):  
Quan Zheng ◽  
Yu Feng Liu
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Unbounded Domain ◽  
Burgers Equation ◽  
High Order ◽  
Computational Domain ◽  
Artificial Boundary ◽  
Artificial Boundary Conditions ◽  
Second Order Convergence ◽  
The Stability

Burgers’ equation on an unbounded domain is an important mathematical model to treat with some external problems of fluid materials. In this paper, we study a FDM of Burgers’ equation using high-order artificial boundary conditions on the unbounded domain. First, the original problem is converted into the heat equation on an unbounded domain by Hopf-Cole transformation. Thus the difficulty of nonlinearity of Burgers’ equation is overcome. Second, high-order artificial boundary conditions are given by using Padé approximation and Laplace transformation. And the conditions confine the heat equation onto a bounded computational domain. Third, we prove the solutions of the resulting heat equation and Burgers’ equation are both stable. Fourth, we establish the FDM for Burgers’ equation on the bounded computational domain. Finally, a numerical example demonstrates the stability, the effectiveness and the second-order convergence of the proposed method.

A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation

2015 ◽  
Vol 8 (4) ◽  
pp. 451-474 ◽  
Author(s):  
Hiu Ning Chan ◽  
Eric T. Chung
Keyword(s):  
Exact Solution ◽  
Numerical Solution ◽  
Discontinuous Galerkin ◽  
Numerical Approximation ◽  
Burgers Equation ◽  
Smooth Solutions ◽  
Regularization Technique ◽  
Tv Regularization ◽  
Inviscid Burgers Equation ◽  
Staggered Discontinuous Galerkin Method

AbstractThe staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approximation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to be taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented.

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