Multilevel numerical solutions of convection-dominated diffusion problems by spline wavelets

2006 ◽  
Vol 22 (4) ◽  
pp. 994-1006 ◽  
Author(s):  
Jiangguo Liu ◽  
Richard E. Ewing ◽  
Guan Qin
Keyword(s):  
Numerical Solutions ◽  
Spline Wavelets ◽  
Convection Dominated ◽  
Diffusion Problems

scholarly journals A Self-Adaptive Numerical Method to Solve Convection-Dominated Diffusion Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Zhi-Wei Cao ◽  
Zhi-Fan Liu ◽  
Zhi-Feng Liu ◽  
Xiao-Hong Wang
Keyword(s):  
Numerical Solutions ◽  
Adaptive Mesh ◽  
Numerical Dissipation ◽  
Flow Problems ◽  
Special Boundary Condition ◽  
Complex Adaptive ◽  
Diffusive Term ◽  
Convection Dominated ◽  
Residual Errors ◽  
Diffusion Problems

Convection-dominated diffusion problems usually develop multiscaled solutions and adaptive mesh is popular to approach high resolution numerical solutions. Most adaptive mesh methods involve complex adaptive operations that not only increase algorithmic complexity but also may introduce numerical dissipation. Hence, it is motivated in this paper to develop an adaptive mesh method which is free from complex adaptive operations. The method is developed based on a range-discrete mesh, which is uniformly distributed in the value domain and has a desirable property of self-adaptivity in the spatial domain. To solve the time-dependent problem, movement of mesh points is tracked according to the governing equation, while their values are fixed. Adaptivity of the mesh points is automatically achieved during the course of solving the discretized equation. Moreover, a singular point resulting from a nonlinear diffusive term can be maintained by treating it as a special boundary condition. Serval numerical tests are performed. Residual errors are found to be independent of the magnitude of diffusive term. The proposed method can serve as a fast and accuracy tool for assessment of propagation of steep fronts in various flow problems.

A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

2013 ◽  
Vol 14 (2) ◽  
pp. 370-392 ◽  
Author(s):  
Eric T. Chung ◽  
Wing Tat Leung
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Heterogeneous Media ◽  
Numerical Solutions ◽  
Solution Space ◽  
Convection Diffusion ◽  
Diffusion And Convection ◽  
Convection Dominated ◽  
Diffusion Problems

AbstractIn this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.

scholarly journals Numerical complete solution for random genetic drift by energetic variational approach

2019 ◽  
Vol 53 (2) ◽  
pp. 615-634 ◽  
Author(s):  
Chenghua Duan ◽  
Chun Liu ◽  
Cheng Wang ◽  
Xingye Yue
Keyword(s):  
Genetic Drift ◽  
Variational Approach ◽  
Numerical Solutions ◽  
Complete Solution ◽  
Random Genetic Drift ◽  
Least Action Principle ◽  
Dirac Delta ◽  
Splitting Technique ◽  
Least Action ◽  
Convection Dominated

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.

scholarly journals Characteristics Weak Galerkin Finite Element Methods for Convection-Dominated Diffusion Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ailing Zhu ◽  
Qiang Xu ◽  
Ziwen Jiang
Keyword(s):  
Finite Element ◽  
Optimal Order ◽  
Weak Galerkin ◽  
Order Error ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Galerkin Finite Element Methods ◽  
Finite Element Procedure ◽  
Convection Dominated ◽  
Diffusion Problems

The weak Galerkin finite element method is combined with the method of characteristics to treat the convection-diffusion problems on the triangular mesh. The optimal order error estimates inH1andL2norms are derived for the corresponding characteristics weak Galerkin finite element procedure. Numerical tests are performed and reported.

Finite element analysis of convection dominated reaction–diffusion problems

2004 ◽  
Vol 48 (2) ◽  
pp. 205-222 ◽  
Author(s):  
A.C. Galeão ◽  
R.C. Almeida ◽  
S.M.C. Malta ◽  
A.F.D. Loula
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Element Analysis ◽  
Convection Dominated ◽  
Diffusion Problems

Robust DPG Method for Convection-Dominated Diffusion Problems

2013 ◽  
Vol 51 (5) ◽  
pp. 2514-2537 ◽  
Author(s):  
Leszek Demkowicz ◽  
Norbert Heuer
Keyword(s):  
Convection Dominated ◽  
Diffusion Problems
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