MULTIGRID HEAT TRANSFER CALCULATIONS USING DIFFERENT ITERATIVE SCHEMES

We propose a multigrid Alternating-Direction Implicit (ADI) method for solving Maxwell's equations in this paper. This method is based on the interpretation of the ADI method as an iterative solver of the Crank-Nicolson (CN) scheme. By introducing a special solving procedure for the residual equation within the ADI framework, multigrid schemes are incorporated into the iterative ADI method. The accuracy and efficiency of the proposed multigrid ADI method are further demonstrated by numerical examples.

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the , where is the order of the Riemann–Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

Stability of ADI schemes applied to convection–diffusion equations with mixed derivative terms

A second order BDF alternating direction implicit difference scheme for the two-dimensional fractional evolution equation

Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions

This paper describes a circuit-oriented leapfrog alternating direction implicit (ADI) method applied to a simulation of a power delivery network (PDN). An equivalent circuit model of the PDN is derived by using a recently-proposed conformal modeling technique, in which orthogonal meshes are used along with a treatment to improve the accuracy of the model. Since the conformal equivalent circuit model has a grid-like topology, it is suitable for direction-based algorithms such as an ADI method and the leapfrog ADI method. An advantage of the ADI-based method is that they can use a relatively-large time step size and reduce the total number of time steps in a transient simulation. In addition, the leapfrog ADI method is more efficient than the existing ADI method because its time marching procedure requires less updating processes. As a result, the conformal equivalent circuit model combined with the leapfrog ADI method can be accurate and efficient in modeling and analysis of the PDN.

Comparison of Implicit Multigrid Schemes for Three-Dimensional Incompressible Flows

Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems

This paper addresses the numerical solution of a fractional evolution equation with a weakly singular kernel. The temporal discretization is carried out by the Crank–Nicolson (CN) method based on graded time steps, while the spatial discretization is carried out using an alternating direction implicit (ADI) finite difference method. Stability and convergence rate are also discussed. The convergence rate is $ O(k^2+h_x^2+h_y^2) $, where $ k $ is a parameter of the maximal time steps and $ h_x $, $ h_y $ are the uniform grid steps in space. Three numerical examples are employed to demonstrate the errors and convergence behavior in practice.

The alternating group explicit (AGE) scheme, together with the alternating direction implicit (ADI) scheme, is used for the parabolic equation (PE) method to analyze the electromagnetic scattering from electrically large conducting targets. The calculation can be taken plane by plane by using the finite difference scheme along the paraxial direction. Furthermore, each transverse plane is divided into two regions, which are a small region around the target and an outer region combined with the air layer and a perfect matching layer. The ADI scheme is applied to the small region, whereas the AGE scheme is introduced to the outer region. In this manner, only a few matrix equations are needed to be solved for the small region and the unknowns of the outer region can be directly obtained without solving any matrix equations. Therefore, the CPU time can be reduced significantly by the proposed method and it can be easily adapted to the parallel technology. The numerical results verify the accuracy and efficiency of the proposed method compared with the Crank-Nicolson-based PE method and the ADI-based PE method. Moreover, the parallel efficiency is given as the last numerical example.

The performance of several approximate factorization methods, coupled with a e nite volume spatial discretization using Roe’ s approximate Riemann solver, are compared by means of numerical tests on a two-dimensional steady inviscid e ow past a blunt body at Mach numbers ranging from 5 to 20. The comparisons are carried out evaluating, by numerical experiments, the optimal Courant number of each method. The alternating direction implicit and the lower ‐upper symmetric Gauss ‐Seidel methods result in the most efe cient factorizations, in terms of CPU time. The former behaves smoothly with increasing Mach number, and its performance is not affected by the grid size. The latter may achieve higher efe ciency but is strongly dependent on the number of relaxation steps performed, requiring optimization in terms of Mach number and grid size. I. Introduction A N IMPLICIT time integration is usually required to improve convergence toward steady state in time-marching, shockcapturing methods. This requirement is particularly true for high Mach number real gas e ows where the practical stability limit of explicit methods is particularly severe. The large block banded linearized operators, resulting from the application of multistep implicitmethodsto multidimensionalstructuredgrids,aresolvedmost efe ciently by resorting to approximate factorization (AF). The implicit operator may be split along the grid directions, as in the early alternating direction implicit (ADI) schemes of Briley and McDonald 1 andBeamand Warming, 2 orintotwolower ‐upper(LU) factors, as in the original LU schemes of Steger and Warming 3 and Jameson and Turkel. 4 The ADI scheme, without the addition of an appropriate amount of numerical damping, may, however, become unstable in three dimensions when formulated in delta form. Therefore, the LU factorization has become increasingly popular, and many improvements to this technique have been proposed in recent years. 5 Yoon and Jameson, 6,7 combined the LU factoriza

Stream-vorticity NS equation ADI scheme to investigate properties of impulsively started translatory flows around the rotational circular cylinder without wall vorticity conditions

Parallelization via domain decomposition techniques of multigrid and adi solvers for navier-stokes equations

Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations