Preconditioned WR–LMF-based method for ODE systems

AbstractThe waveform relaxation (WR) method was developed as an iterative method for solving large systems of ordinary differential equations (ODEs). In each WR iteration, we are required to solve a system of ODEs. We then introduce the boundary value method (BVM) which is a relatively new method based on the linear multistep formulae to solve ODEs. In particular, we apply the generalized minimal residual method with the Strang-type block-circulant preconditioner for solving linear systems arising from the application of BVMs to each WR iteration. It is demonstrated that these techniques are very effective in speeding up the convergence rate of the resulting iterative processes. Numerical experiments are presented to illustrate the effectiveness of our methods

Circulant preconditioners for solving differential equations with multidelays

AbstractWe consider the solution of differential equations with multidelays by using boundary value methods (BVMs). These methods require the solution of some nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed to solve these linear systems. If an Ak1,k2-stable BVM is used, we show that our preconditioner is invertible and the spectrum of the preconditioned matrix is clustered. It follows that when the GMRES method is applied to solving the preconditioned systems, the method would converge fast. Numerical results are given to show the effectiveness of our methods

Strang-type Preconditioners for Solving Linear Systems from Neutral Delay Di®erential Equations

We study the solution of neutral delay di®erential equations (NDDEs) by using boundary value methods (BVMs). The BVMs require the solution of nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed to solve these linear systems. We show that if an Ak1;k2-stable BVM is used for solving an m-by-m system of NDDEs, then our preconditioner is invertible and the spectrum of the preconditioned system is clustered. It follows that when the GMRES method is applied to the preconditioned systems, the method could converge fast. Numerical results are given to show that our method is e®ective

Strang-type Preconditioners for Solving System of Delay Differential Equations by Boundary Value Methods

In this paper,we survey some of the latest developments in using boundary value methods (BVMs) for solving systems of delay differential equations (DDEs) These methods require the solutions of nonsymmetric,large and sparse linear systems. The GMRES method with the Strang-type preconditioner is proposed for solving these systems. One of the main results is that if an Av1,v2- stable BVM is used for a system of DDEs, then the preconditioner is invertible and the preconditioned matrix can be decomposed as I + L where I is the identity matrix and L is a low rank matrix.It follows that when the GMRES method is applied to solving the preconditioned systems,the method will converge fast

Efficient preconditioning for sequences of parametric complex symmetric linear systems

Solution of sequences of complex symmetric linear systems of the form Ajxj = bj, j = 0,..., s, Aj = A + αjEj, A Hermitian, E0, ..., E a complex diagonal matrices and α0, ..., αa scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent PDEs; lattice gauge computations in quantum chromodynamics; the Helmholtz equation; shift-and-invert and Jacobi-Davidson algorithms for large-scale eigenvalue calculations; problems in control theory and many others. If A is symmetric and has real entries then Aj is complex symmetric. The case A Hermitian positive semideflnite, Re(αj) ≥ 0 and such that the diagonal entries of E j, j = 0,..., s have nonnegative real part is considered here. Some strategies based on the update of incomplete factorizations of the matrix A and A-1 are introduced and analyzed. The numerical solution of sequences of algebraic linear systems from the discretization of the real and complex Helmholtz equation and of the diffusion equation in a rectangle illustrate the performance of the proposed approaches