scholarly journals A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection–Diffusion Problems with Boundary/Interior Layers

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1123
Author(s):  
Tianlong Ma ◽  
Lin Zhang ◽  
Fujun Cao ◽  
Yongbin Ge
Keyword(s):  
Present Method ◽  
Multigrid Method ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Finite Difference Schemes ◽  
Interior Layer ◽  
Convection Diffusion ◽  
Uniform Grids ◽  
Interior Layers ◽  
Diffusion Problems

Boundary or interior layer problems of high-dimensional convection–diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric problems. In this paper, we extended our previous work on the partial semi-coarsening multigrid method combined with the high-order compact (HOC) difference scheme for solving the two-dimensional (2D) convection–diffusion problems on non-uniform grids to the three-dimensional (3D) cases. The main merit of the present method is that the multigrid method on non-uniform grids can be performed with a different number of grids in different coordinate axes, which is more efficient than the multigrid method on non-uniform grids with the same number of grids in different coordinate axes. Numerical experiments are carried out to validate the accuracy and efficiency of the present method. It is shown that, without losing the high precision, the present method is very effective to reduce computing cost by cutting down the number of grids in the direction(s) which does/do not contain boundary or interior layer(s).

Numerical solutions of singularly perturbed convection-diffusion problems

2014 ◽  
Vol 24 (6) ◽  
pp. 1268-1274 ◽  
Author(s):  
Fazhan Geng ◽  
Suping Qian ◽  
Shuai Li
Keyword(s):  
Asymptotic Expansion ◽  
Present Method ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Content Type ◽  
Convection Diffusion ◽  
Asymptotic Expansion Method ◽  
Diffusion Problems

Purpose – The purpose of this paper is to find an effective numerical method for solving singularly perturbed convection-diffusion problems. Design/methodology/approach – The present method is based on the asymptotic expansion method and the variational iteration method (VIM). First a zeroth order asymptotic expansion for the solution of the given singularly perturbed convection-diffusion problem is constructed. Then the reduced terminal value problem is solved by using the VIM. Findings – Two numerical examples are introduced to show the validity of the present method. Obtained numerical results show that the present method can provide very accurate analytical approximate solutions not only in the boundary layer, but also away from the layer. Originality/value – The combination of the asymptotic expansion method and the VIM is applied to singularly perturbed convection-diffusion problems.

Chebyshev multidomain pseudospectral method to solve cardiac wave equations with rotational anisotropy

2018 ◽  
Vol 09 (04) ◽  
pp. 1850025 ◽  
Author(s):  
Jairo Rodríguez-Padilla ◽  
Daniel Olmos-Liceaga
Keyword(s):  
Wave Propagation ◽  
Numerical Methods ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Pseudospectral Method ◽  
Finite Difference Schemes ◽  
Cardiac Models ◽  
Computational Resources ◽  
Realistic Geometries

The implementation of numerical methods to solve and study equations for cardiac wave propagation in realistic geometries is very costly, in terms of computational resources. The aim of this work is to show the improvement that can be obtained with Chebyshev polynomials-based methods over the classical finite difference schemes to obtain numerical solutions of cardiac models. To this end, we present a Chebyshev multidomain (CMD) Pseudospectral method to solve a simple two variable cardiac models on three-dimensional anisotropic media and we show the usefulness of the method over the traditional finite differences scheme widely used in the literature.

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

2004 ◽  
Vol 4 (1) ◽  
pp. 105-127 ◽  
Author(s):  
Grigory I. Shishkin ◽  
Lidia P. Shishkina ◽  
Pieter W. Hemker
Keyword(s):  
Coordinate System ◽  
Transition Layer ◽  
A Priori ◽  
Adaptive Mesh ◽  
Singularly Perturbed ◽  
Finite Difference Schemes ◽  
A Posteriori ◽  
Interior Layer ◽  
Convection Diffusion ◽  
Mesh Technique

AbstractWe study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’.

Calculation of External and Internal Transonic Flow Field of a Three-Dimensional S-Shaped Inlet

1985 ◽  
Author(s):  
Shen Huili ◽  
Luo Shijun ◽  
Ji Minggang ◽  
Xing Zongwen ◽  
Zhu Xin ◽  
...  
Keyword(s):  
Finite Difference ◽  
Flow Field ◽  
Transonic Flow ◽  
Present Method ◽  
Velocity Potential ◽  
Free Stream ◽  
Three Dimensional ◽  
Finite Difference Schemes ◽  
Finite Difference Equations ◽  
Free Stream Mach Number

A mixed finite difference method for calculating the external and internal transonic flow field around an s-shaped inlet is presented. Starting from the velocity potential equation and using Cartesian mesh and mixed finite difference schemes, the authors have obtained a system of finite difference equations and solved them with the aid of alternating line relaxations along two directions. Computations have been made for an s-shaped inlet with free stream Mach number M=0.8 at different angles of attack. Computed results are compared with those computed by perturbation method and with experimental results. Such a comparison shows that the present method is promising.

scholarly journals Singularly perturbed convection–diffusion problems with boundary and weak interior layers

2004 ◽  
Vol 166 (1) ◽  
pp. 133-151 ◽  
Author(s):  
P.A. Farrell ◽  
A.F. Hegarty ◽  
J.J.H. Miller ◽  
E. O'Riordan ◽  
G.I. Shishkin
Keyword(s):  
Singularly Perturbed ◽  
Convection Diffusion ◽  
Interior Layers ◽  
Diffusion Problems
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