scholarly journals A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

2021 ◽  
Vol 29 (1) ◽  
pp. 1819-1839
Author(s):  
Guoliang Zhang ◽  
◽  
Shaoqin Zheng ◽  
Tao Xiong ◽  
Keyword(s):  
Finite Difference ◽  
Hyperbolic Equations ◽  
Weno Scheme ◽  
Nonlinear Hyperbolic Equations ◽  
One Dimensional ◽  
Exponential Integrator

Computing experience with hyperbolic partial differential equations

1971 ◽  
Vol 323 (1553) ◽  
pp. 263-270 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Hyperbolic Equations ◽  
Artificial Viscosity ◽  
Hyperbolic Partial Differential Equations ◽  
Nonlinear Hyperbolic Equations ◽  
Particle In Cell ◽  
Large Numbers ◽  
Eulerian Mesh

Three methods of integrating nonlinear hyperbolic equations are considered, namely the characteristic, mesh finite-difference and particle-in-cell techniques. In particular, a practical characteristic code for problems with large numbers of discontinuities is described, and developments in the use of artificial viscosity terms are outlined. The incorporation of certain particle-in-cell features into Eulerian mesh methods is also illustrated.

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.

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.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

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