The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

Abstract

Abstract. We implement an exponential integrator for large and sparse systems of ODEs, generated by FE (Finite Element) discretization with mass-lumping of advection-diffusion equations. The relevant exponentiallike matrix function is approximated by polynomial interpolation, at a sequence of real Leja points related to the spectrum of the FE matrix (ReLPM, Real Leja Points Method). Application to 2D and 3D advection-dispersion models shows speed-ups of one order of magnitude with respect to a classical variable step-size Crank-Nicolson solver. The Advection-Diffusion ModelWe consider the classical evolutionary advection-diffusion problemwith mixed Dirichlet and Neumann boundary conditions on. Equation (1) represents, e.g., a simplified model for solute transport in groundwater flow (advection-dispersion), where c is the solute concentration, D the hydrodynamic dispersion tensor, v the average linear velocity of groundwater flow andWork supported by the research project CPDA028291 "Efficient approximation methods for nonlocal discrete transforms" of the University of Padova, and by the subproject "Approximation of matrix functions in the numerical solution of differential equations" (co-ordinator M. Vianello, University of Padova) of the MIUR PRIN 2003 project "Dynamical systems on matrix manifolds: numerical methods and applications" (co-ordinator L. Lopez, University of Bari). Thanks also to the numerical analysis group at the Dept. of Math. Methods and Models for Appl. Sciences of the University of Padova, for having provided FE matrices for our numerical tests, and for the use of their computing resources.

Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise

“…Here we focus on the real fast Léja points and the Krylov subspace techniques to evaluate the action of the exponential matrix function ϕ i (∆t A h ) on a vector v, instead of computing the full exponential function ϕ i (∆t A h ) as in a standard Padé approximation. The details of the real fast Léja points technique and [33,34,17] for the Krylov subspace technique are given in [39,31,32]. We give a brief summary below.…”

“…The cost of the extra accuracy though is that to implement these methods we need to compute the exponential of a non-diagonal matrix, which is a notorious problem in numerical analysis [35]. However, new developments for both Léja points and Krylov subspace techniques [33,34,17,39,31,32] have led to efficient methods for computing matrix exponentials. Compared to the Fourier-Galerkin methods of [2,3,4,7] we gain the flexibility of finite element (or finite volume methods) to deal with complex boundary conditions and we can apply well developed techniques such as upwinding to deal with advection.…”

Comparing Leja and Krylov Approximations of Large Scale Matrix Exponentials

“…In the FE case it is obtained cheaply by left-multiplying the stiffness matrix with the inverse of a diagonal mass matrix, via the mass-lumping technique (cf. [2]). …”

“…Among others, the ReLPM (Real Leja Points Method), proposed in [4] and applied to advection-diffusion models in [2], has shown very attractive computational features. It rests on Newton interpolation of the exponential functions at a sequence of Leja points on the real focal interval of a family of confocal ellipses in the complex plane.…”

Approximation of matrix operators applied to multiple vectors

“…The most popular method for approximating such matrix operators is by Krylov subspace methods [2][3][4][11][12][13]15,16,19,23,24,32,34,36]. A more recent development for symmetric (or sectorial) matrices is to use rational approximations [26,29,30,37,38].…”