A Fast Semi-implicit Method for Anisotropic Diffusion

Simple finite differencing of the anisotropic diffusion equation, where diffusion is only along a given direction, does not ensure that the numerically calculated heat fluxes are in the correct direction. This can lead to negative temperatures for the anisotropic thermal diffusion equation. In a previous paper we proposed a monotonicity-preserving explicit method which uses limiters (analogous to those used in the solution of hyperbolic equations) to interpolate the temperature gradients at cell faces. However, being explicit, this method was limited by a restrictive Courant-Friedrichs-Lewy (CFL) stability timestep. Here we propose a fast, conservative, directionally-split, semi-implicit method which is second order accurate in space, is stable for large timesteps, and is easy to implement in parallel. Although not strictly monotonicity-preserving, our method gives only small amplitude temperature oscillations at large temperature gradients, and the oscillations are damped in time. With numerical experiments we show that our semi-implicit method can achieve large speed-ups compared to the explicit method, without seriously violating the monotonicity constraint. This method can also be applied to isotropic diffusion, both on regular and distorted meshes.Comment: accepted in the Journal of Computational Physics; 13 pages, 7 figures; updated to the accepted versio

Some observations on weighted GMRES

We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present new alternative implementations of the weighted Arnoldi algorithm which may be favorable in terms of computational complexity, and examine stability issues connected with these implementations. Two implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used

Grid computing: a case study in hybrid GMRES method

Abstract. Grid computing in general is a special type of parallel computing. It intends to deliver high-performance computing over distributed platforms for computation and data-intensive applications by making use of a very large amount of resources. The GMRES method is used widely to solve the large sparse linear systems. In this paper, we present an effective parallel hybrid asynchronous method, which combines the typical parallel GMRES method with the Least Square method that needs some eigenvalues obtained from a parallel Arnoldi process. And we apply it on a Grid Computing platform Grid5000. From the numeric results, we will present that this hybrid method has some advantage for some real or complex systems compared to the general method GMRES

A framework for deflated and augmented Krylov subspace methods

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.Comment: 24 pages, 3 figure

A point collocation method for geometrically nonlinear membranes

AbstractThis paper describes the development of a numerical model for geometrically nonlinear membranes and evaluates its performance for membranes at static equilibrium. The scheme has several features not commonly seen in structural finite element analysis: the point collocation method, group formulation, and a staggered mesh. In the point collocation finite element method, the partial differential equations are solved at each node instead of by integrating over elements. The group formulation simplifies the handling of nonlinearities by interpolating the nonlinear products of variables, as opposed to seeking the product of independently interpolated variables. The domain is discretized with a staggered mesh of linear triangles and associated polygons. Two sequential gradient recovery operations are performed: first the gradients of the linear triangles are calculated and converted to stresses; then, polygon derivative shape functions derived in this paper are used to determine the internal forces from the stress gradients. The resulting system of nonlinear equations is solved with a Jacobian-free Newton–Krylov solver. The code is first verified using the patch test and the method of manufactured solutions. Then the results are validated using experimental data and benchmark code results in the literature for the Hencky problem (a circular membrane with a fixed perimeter and uniform inflation pressure). The observed rates of convergence for both displacement and radial strain were two. For the configurations and grids used in this investigation, the scheme was suitable for accurately predicting sub-hyperelastic deformations