An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions

An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions

It is very natural to solve time dependent problems with Domain Decomposition Methods by using an implicit scheme for the time variable and then applying a classical iterative domain decomposition method at each time step. This is however not what the Schwarz Waveform Relaxation (SWR) methods do. The SWR methods are a combination of the Schwarz Domain Decomposition methods, see [10], and the Waveform Relaxation algorithm, see [7]. Combined, one obtains a new method which decomposes the domain into subdomains on which time dependent problems are solved. Iterations are then introduced, where communication between subdomains is done at artificial interfaces along the whole time window.KeywordsMaxwell EquationDomain DecompositionDomain Decomposition MethodTime Dependent ProblemGlobal DomainThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, the streamline diffusion weak Galerkin finite element method is proposed and analyzed for solving unsteady time convection diffusion problem in two dimension. The v-elliptic property and the stability of this scheme are proved in terms of some conditions. We derive an error estimate in L2(μ) and H1(μ) norm. Numerical experiments have demonstrated the effectiveness of the method in solving convection propagation problems, and the theoretical analysis has been validated.

This paper is devoted to the derivation of a pleasingly parallel Galerkin method for the time-independent N-body Schrodinger equation, and its time-dependent version modeling molecules subject to an external electric field (Bandrauk 1994; Bandrauk et al., J. Phys. B-Atom. Mol. Opt. Phys. 46(15), 153001, 2013; Cohen-Tannoudji et al. 1992). In this goal, we develop a Schwarz waveform relaxation (SWR) domain decomposition method (DDM) for the N-body Schrodinger equation. In order to optimize the efficiency and accuracy of the overall algorithm, (i) we use mollifiers to regularize the singular potentials and to approximate the Schrodinger Hamiltonian, (ii) we select appropriate orbitals, and (iii) we carefully derive and approximate the SWR transmission conditions. Some low-dimensional numerical experiments are presented to illustrate the methodology.

In practical engineering applications, convection diffusion equations are generally used to describe the transport processes involving fluid motion, heat transfer, astrophysics, oceanography, meteorology, semiconductors, hydraulics, pollutant & sediment transport and chemical engineering. In this paper, a high order compact difference scheme based on the fourth order compact difference scheme in spatial discretization and the fourth order Runge-Kutta method in time integration is proposed for the numerical simulation of the unsteady convection-diffusion equation. The validity and effectiveness of the proposed method is firstly tested by a two-dimensional convection-diffusion equation with a Gaussian pulse type concentration. The L 2 error norms are used to measure differences between the exact and numerical solutions and compared to those obtained by other methods. It is shown that the results obtained by proposed method agree very well with the analytical solutions and is more accurate than other methods. Then, a two-dimensional non-linear Burgers equation is used to validate the effectiveness of the proposed method used to solve the non-linear convection-diffusion equation, which also models well. Finally, the Taylor’s vortex problem is investigated by the proposed method and good agreement is obtained with the exact solutions. From the three test problems, it is shown that the proposed high order compact difference scheme is an efficient and accurate method to simulate the transport problems and also can be applied to many engineering problems.

We introduce a nonoverlapping variant of the Schwarz waveform relaxation algorithm for semilinear wave propagation in one dimension. Using the theory of absorbing boundary conditions, we derive a new nonlinear algorithm. We show that the algorithm is well-posed and we prove its convergence by energy estimates and a Galerkin method. We then introduce an explicit scheme. We prove the convergence of the discrete algorithm with suitable assumptions on the nonlinearity. We finally illustrate our analysis with numerical experiments.

The objective of this paper is to describe a parallel acceleration framework of the additive schwarz waveform relaxation for parabolic problems. The problem is in three space dimension and time. This new parallel domain decomposition algorithm generalizes the Aitken-like Acceleration method of the additive Schwarz algorithm for elliptic problems. Although the standard Schwarz Waveform Relaxation algorithm has a linear rate of convergence and low numerical efficiency, it is beneficent to cache use and scales with the memory. The combination with the Aitken-like Acceleration method transforms the Schwarz algorithm into a direct solver for the heat operator. This solver combines all in one the load balancing, the efficiency, the scalability and the fault tolerance features which make it suitable for grid environments.KeywordsDomain DecompositionParabolic ProblemSolver ProcessDomain Decomposition MethodInterface ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Convergence analysis of Schwarz waveform relaxation method to compute coupled advection–diffusion–reaction equations

In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

Hydrodynamic dispersion of reactive solute in a Hagen–Poiseuille flow of a layered liquid

Nodal Integral method for multi-group neutron diffusion equation in three dimensional cylindrical coordinate system

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Péclet number 0 ≤ Pe ≤ 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.

A high order finite difference scheme is introduced to deal with thermo-hydraulic quench simulation in superconducting magnets, specifically in the case of cable-in-conduit conductors (CICC). Liquid helium flow in CICC is governed by the unsteady convection-diffusion conservation equations, in the low Mach number and high Reynolds number flow region. In the past several years many methods have been considered for quench simulation in superconducting magnets, for example, finite element, finite volume, and finite difference methods. One of the main challenges is the accurate simulation of flow during a superconductor quench, given the large gradients in pressure and temperature that arise during conductor normalization. These simulations are key to proper magnet protection design. The Dispersion Relation Preserving (DRP) scheme is applied to the quench simulation problem. The computational results are compared to those obtained using a well-proven commercial quench simulation code (Gandalf), as well as experimental data. Preliminary results are promising.

The effect of magnetic field on unsteady convective diffusion in a couple stress fluid (blood) is studied using a time dependent dispersion model. This model is used to calculate the mean concentration distribution of a solute, bounded by the porous layer and is expressed as a function of dimensionless axial distance and time. The magnetic field, arising as a body couple in the governing equations is shown to increase the axis dispersion coefficient. This is useful to the control of haemolysis caused by artificial organs implanted or extracorporeal. Dispersion coefficient and mean concentration are computed for different values of Hartmann number (M), Couple Stress Parameter (a) and Porous Parameter (σ).