scholarly journals A Triangular Spectral Element Method for the 2-D Viscous Burgers Equation

2022 ◽  
Vol 2148 (1) ◽  
pp. 012014
Author(s):  
Mengyu Zhang ◽  
Hua Wu
Keyword(s):  
Spectral Element Method ◽  
Nonlinear Term ◽  
Burgers Equation ◽  
Implementation Process ◽  
Spectral Element ◽  
Basis Functions ◽  
Spatial Direction ◽  
Fully Discrete ◽  
New Type ◽  
Element Method

Abstract A triangular spectral element method is established for the two-dimensional viscous Burgers equation. In the spatial direction, a new type of mapping is applied. We splice the local basis functions on each triangle into a global basis function. The second-order Crank-Nicolson/ leap-frog (CNLF) method is used for discretization in the time direction. Due to the use of a quasi-interpolation operator, the nonlinear term can be handled conveniently. We give the fully discrete scheme of the method and the implementation process of the algorithm. Numerical examples verify the effectiveness of this method.

A Space-Time Discontinuous Galerkin Spectral Element Method for Nonlinear Hyperbolic Problems

2018 ◽  
Vol 16 (01) ◽  
pp. 1850093 ◽  
Author(s):  
Chaoxu Pei ◽  
Mark Sussman ◽  
M. Yousuff Hussaini
Keyword(s):  
Discontinuous Galerkin ◽  
Spectral Element Method ◽  
Burgers Equation ◽  
Spectral Element ◽  
Space Time ◽  
Picard Iteration ◽  
Spectral Accuracy ◽  
Element Discretization ◽  
Space And Time ◽  
Element Method

A space-time discontinuous Galerkin spectral element method is combined with two different approaches for treating problems with discontinuous solutions: (i) adding a space-time dependent artificial viscosity, and (ii) tracking the discontinuity with space-time spectral accuracy. A Picard iteration method is employed to solve nonlinear system of equations derived from the space-time DG spectral element discretization. Spectral accuracy in both space and time is demonstrated for the Burgers’ equation with a smooth solution. For tests with discontinuities, the present space-time method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. The spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers’ equation obtained by the tracking method.

Wavelet Element Method for Computing Band Structures of Phononic Crystals

2013 ◽  
Author(s):  
Zhangyi Liu ◽  
Jiu Hui Wu
Keyword(s):  
Boundary Conditions ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Structural Features ◽  
Phononic Crystals ◽  
Basis Functions ◽  
Bounded Domains ◽  
Biorthogonal Wavelet ◽  
First Order ◽  
Element Method

In this paper we combine biorthogonal wavelet systems with the philosophy of Spectral Element Method to obtain a biorthogonal wavelet system on fairly general bounded domains. We also extend the boundary adaption of wavelet elements to first order derivatives allowing the construction of basis functions that exactly satisfy boundary conditions. Since this method allows us to take advantage of structural features of phononic crystals and the boundary conditions are satisfied rigorously, a better accuracy and higher efficiency can be obtained.

A Hybrid Spectral Element Method for Fractional Two-Point Boundary Value Problems

2017 ◽  
Vol 10 (2) ◽  
pp. 437-464 ◽  
Author(s):  
Changtao Sheng ◽  
Jie Shen
Keyword(s):  
Fractional Derivatives ◽  
Spectral Element Method ◽  
Boundary Value ◽  
Spectral Element ◽  
Basis Functions ◽  
Point Boundary ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Jacobi Functions ◽  
Element Method

AbstractWe propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions att= 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates underL2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

3-D dc resistivity modelling based on spectral element method with unstructured tetrahedral grids

2019 ◽  
Vol 220 (3) ◽  
pp. 1748-1761
Author(s):  
Jiao Zhu ◽  
Changchun Yin ◽  
Youshan Liu ◽  
Yunhe Liu ◽  
Ling Liu ◽  
...  
Keyword(s):  
Spectral Element Method ◽  
Unstructured Grids ◽  
Spectral Element ◽  
Basis Functions ◽  
Earth Model ◽  
Dc Resistivity ◽  
Underground Structures ◽  
Tetrahedral Grids ◽  
Interpolation Polynomials ◽  
Element Method

SUMMARY In this paper, we propose a spectral element method (SEM) based on unstructured tetrahedral grids for direct current (dc) resistivity modelling. Unlike the tensor-product of 1-D Gauss–Lobatto–Legendre (GLL) quadrature in conventional SEM, we use Proriol–Koornwinder–Dubiner (PKD) polynomials to form the high-order basis polynomials on tetrahedral grids. The final basis functions are established by using Vandermonde matrix. Compared to traditional SEM, our method truly takes into account the high precision of spectral method and the flexibility of finite element method with unstructured grids for modelling the complex underground structures. After addressing the theory on the construction of basis functions and interpolation and integration nodes, we validate our algorithm using the analytical solutions for a layered earth model and the results from other methods for multiple geoelectrical models. We further investigate a dual-track scheme for improving the accuracy of our SEM by increasing the order of interpolation polynomials or by refining the grids.

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