scholarly journals Fifth-Order Mapped Semi-Lagrangian Weighted Essentially Nonoscillatory Methods Near Certain Smooth Extrema

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Lang Wu ◽  
Dazhi Zhang ◽  
Boying Wu ◽  
Xiong Meng
Keyword(s):  
Finite Difference ◽  
Finite Volume ◽  
Interpolation Scheme ◽  
Numerical Examples ◽  
Weighted Essentially Nonoscillatory ◽  
Accuracy Test ◽  
A Cell ◽  
Fifth Order ◽  
Cell Average ◽  
Essentially Nonoscillatory

Fifth-order mapped semi-Lagrangian weighted essentially nonoscillatory (WENO) methods at certain smooth extrema are developed in this study. The schemes contain the mapped semi-Lagrangian finite volume (M-SL-FV) WENO 5 method and the mapped compact semi-Lagrangian finite difference (M-C-SL-FD) WENO 5 method. The weights in the more common scheme lose accuracy at certain smooth extrema. We introduce mapped weighting to handle the problem. In general, a cell average is applied to construct the M-SL-FV WENO 5 reconstruction, and the M-C-SL-FD WENO 5 interpolation scheme is proposed based on an interpolation approach. An accuracy test and numerical examples are used to demonstrate that the two schemes reduce the loss of accuracy and improve the ability to capture discontinuities.

A new type of finite difference WENO schemes for Hamilton–Jacobi equations

2019 ◽  
Vol 30 (02n03) ◽  
pp. 1950020 ◽  
Author(s):  
Xiaohan Cheng ◽  
Jianhu Feng ◽  
Supei Zheng ◽  
Xueli Song
Keyword(s):  
Finite Difference ◽  
Weno Schemes ◽  
Numerical Examples ◽  
Weighted Essentially Nonoscillatory ◽  
New Type ◽  
Spurious Oscillations ◽  
Symmetry Requirement ◽  
Finite Difference Weno Schemes ◽  
Jacobi Equations ◽  
Fifth Order

In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.

scholarly journals The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Changna Lu ◽  
Luoyan Xie ◽  
Hongwei Yang
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Time Discretization ◽  
Weno Scheme ◽  
Weighted Essentially Nonoscillatory ◽  
Accuracy Order ◽  
Essentially Nonoscillatory

A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.

scholarly journals An Optimized Symmetric WENO Method-Based Numerical Simulation of Intense Sound Field Generated by Underwater Plasma Sound Source

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Kaizhuo Lei ◽  
Xiaolong Liu ◽  
Ning Li ◽  
Xuchao Fan
Keyword(s):  
Euler Equations ◽  
Sound Source ◽  
Sound Field ◽  
Maximum Error ◽  
Numerical Examples ◽  
Weighted Essentially Nonoscillatory ◽  
The Third ◽  
Calculation Results ◽  
Pulse Sound ◽  
Fifth Order

The intensive pulse sound wave can be generated by the underwater plasma sound source (UPSS) based on the discharge of the underwater high voltage. The distribution of the sound field is prominently nonlinear. In this paper, the sound field of the intensive UPSS is described by the integral two-dimensional axisymmetric unsteady Euler equations firstly. In order to solve the Euler equations numerically, an optimized fifth-order symmetric WENO (weighted essentially nonoscillatory) method based on the three templates is proposed which is called WENO-SYM3. Without increasing the number of candidate templates, a new symmetric template structure can be obtained by expanding the second template and shifting the third one backwards for one space. The method is validated through numerical examples and experiments, and the results show that WENO-SYM3 has a high distinguished accuracy; meanwhile, its nonphysical oscillations are not obvious. The experimental results are basically the same as the calculation results, and the maximum error is around 3%.

scholarly journals A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

2018 ◽  
Vol 40 (1) ◽  
pp. 405-421 ◽  
Author(s):  
N Chatterjee ◽  
U S Fjordholm
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Numerical Examples ◽  
Continuity Equations ◽  
Multiple Dimensions ◽  
Nonlocal Conservation Laws ◽  
Volume Method ◽  
Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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