Should exponential integrators be used for advection-dominated problems?

Abstract

In this paper, we consider the application of exponential integrators to problems that are advection dominated, either on the entire or on a subset of the domain. In this context, we compare Leja and Krylov based methods to compute the action of exponential and related matrix functions. We set up a performance model by counting the different operations needed to implement the considered algorithms. This model assumes that the evaluation of the right-hand side is memory bound and allows us to evaluate performance in a hardware independent way. We find that exponential integrators perform comparably to explicit Runge-Kutta schemes for problems that are advection dominated in the entire domain. Moreover, they are able to outperform explicit methods in situations where small parts of the domain are diffusion dominated. We generally observe that Leja based methods outperform Krylov iterations in the problems considered. This is in particular true if computing inner products is expensive.

Figures (7)

• Figure 1: Spectrum of the matrix $-(A_h + B_h)$ for the linear problem \ref{['problem1']} for different choices of $\kappa$. In the left figure, the choice $\kappa = 1/80$ indicates a diffusion-dominated problem, while the choice $\kappa = 1/2560$ indicates an advection-dominated problem. The figure on the right, where the function $\kappa$ is defined as described in Section \ref{['mixproblem']}, shows the spectrum of a mixed problem which, depending on the position, is either diffusion- or advection-dominated.
• Figure 2: The spectra of the Jacobians evaluated at the initial value (left figure) and at the final value (right figure) of a numerical solution of the two-dimensional compressible isothermal Navier--Stokes problem \ref{['eq41']} with diffusion coefficients of $\nu = 10^{-4}$ and $\nu = 10^{-6}$, respectively, indicating advection-dominated scenarios.
• Figure 3: The numerical results for the diffusion-dominated problem \ref{['problem1']} with $\kappa=\frac{1}{80}$ are presented in the upper two panes. The figures illustrate the achieved accuracy and computational cost of the considered methods for evaluating $\exp(- (A_h+B_h))u_0$ as a function of the time step size. The tolerance is chosen such that the achieved accuracy for the exponential integrators is $10^{-4}$ (dashed-dotted line) and $10^{-7}$ (solid line), respectively. The lower two panes depict the achieved accuracy of the considered methods as a function of the computational cost for two chosen values of $\tau$ and $\zeta$.
• Figure 4: The numerical results for the advection-dominated problem \ref{['problem1']} with $\kappa=\frac{1}{2560}$ are shown in the upper two panes. The figures illustrate the achieved accuracy and computational cost of the considered methods for evaluating $\exp(- (A_h+B_h))u_0$ as a function of the time step size. The tolerance is chosen such that the achieved accuracy for the exponential integrators is $10^{-4}$ (dashed-dotted line) and $10^{-7}$ (solid line), respectively. The lower two panes depict the achieved accuracy of the considered methods as a function of the computational cost for two different values of $\tau$ and $\zeta$.
• Figure 5: The numerical results for the mixed problem, as defined in equation \ref{['problem1']} and \ref{['eq12']} are presented in the upper two panes. The figures illustrate the achieved accuracy and the required computational cost of the considered methods for evaluating $\exp(- (A_h+B_h))u_0$ as a function of the time step size. The tolerance is chosen such that the achieved accuracy for the exponential integrators is $10^{-4}$ (dashed-dotted line) and $10^{-7}$ (solid line), respectively. The pane \ref{['fig4cc']} illustrates the function $\kappa$ in \ref{['eq12']}. The pane \ref{['fig4d']} depicts the achieved accuracy of the considered methods as a function of the computational cost for two different values of $\tau$ and $\zeta$. For Krylov methods the employed time step size is $\tau = \frac{1}{80}$.