scholarly journals Suppressing Numerical Oscillation for Nonlinear Hyperbolic Equations by Wavelet Analysis

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Yong Zhao ◽  
Peng-Yao Yu ◽  
Shao-Juan Su ◽  
Tian-Lin Wang
Keyword(s):  
Shock Wave ◽  
Wavelet Analysis ◽  
Numerical Solution ◽  
Hyperbolic Equations ◽  
Real Solution ◽  
Nonlinear Hyperbolic Equations ◽  
Wavelet Shrinkage ◽  
One Dimensional ◽  
The Real ◽  
Dual Wavelet

In the numerical solution for nonlinear hyperbolic equations, numerical oscillation often shows and hides the real solution with the progress of computation. Using wavelet analysis, a dual wavelet shrinkage procedure is proposed, which allows one to extract the real solution hidden in the numerical solution with oscillation. The dual wavelet shrinkage procedure is introduced after applying the local differential quadrature method, which is a straightforward technique to calculate the spatial derivatives. Results free from numerical oscillation can be obtained, which can not only capture the position of shock and rarefaction waves, but also keep the sharp gradient structure within the shock wave. Three model problems—a one-dimensional dam-break flow governed by shallow water equations, and the propagation of a one-dimensional and a two-dimensional shock wave controlled by the Euler equations—are used to confirm the validity of the proposed procedure.

The formation of gas hydrates under shock wave impact on the bubble media (two-dimensional case)

2010 ◽  
Vol 7 ◽  
pp. 90-97
Author(s):  
M.N. Galimzianov ◽  
I.A. Chiglintsev ◽  
U.O. Agisheva ◽  
V.A. Buzina
Keyword(s):  
Shock Wave ◽  
Gas Hydrates ◽  
Hydrate Formation ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Wave Impact ◽  
Bubble Liquid ◽  
One Dimensional ◽  
Bubble Media ◽  
Main Factors

Formation of gas hydrates under shock wave impact on bubble media (two-dimensional case) The dynamics of plane one-dimensional shock waves applied to the available experimental data for the water–freon media is studied on the base of the theoretical model of the bubble liquid improved with taking into account possible hydrate formation. The scheme of accounting of the bubble crushing in a shock wave that is one of the main factors in the hydrate formation intensification with increasing shock wave amplitude is proposed.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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