From Memory Model to CPU Time: Exponential Integrators for Advection-Dominated Problems

TL;DR

The paper evaluates exponential integrators—specifically Krylov subspace and Leja interpolation—for advection-dominated PDEs, using exponential Rosenbrock schemes to approximate matrix functions. By applying these methods to linear advection–diffusion and nonlinear 2D compressible isothermal Navier–Stokes problems, it shows that Leja-based approaches excel with large time steps, while Krylov methods perform best at smaller steps, with both often surpassing explicit Runge–Kutta schemes in efficiency and accuracy. The study includes a linear regime, a strongly advection-dominated regime, and mixed regimes, plus nonlinear explosion and shear-flow tests, providing a comprehensive comparison of CPU-time and error (work–precision) across regimes. Overall, exponential integrators offer practical, general-purpose advantages for advection-dominated problems, particularly in linear and mixed domains, though nonlinear cases can narrow the performance gap depending on tolerances and step sizes.

Abstract

In this paper, we investigate the application of exponential integrators to advection-dominated problems. We focus on Krylov subspace and Leja interpolation methods to compute the action of exponential and related matrix functions. Complementing our earlier paper, arXiv:2410.12765 (to appear in Advances in Applied Mathematics and Mechanics, 2025) based on a performance model, we extend the numerical investigation to higher-order Krylov approximations and new numerical regime, and assess their CPU-time efficiency relative to explicit Runge--Kutta schemes. We show that, depending on the problem setting, exponential integrators can either outperform or match explicit Runge--Kutta schemes. We also observe that Leja-based methods outperform Krylov iterations for large time steps, whereas for small time steps, Krylov-based methods provide better results than Leja-based methods.

Figures (11)

• Figure 1: Numerical spectrum of the matrix $-(A_h + B_h)$ for problem \ref{['problem1']} for different values of $\kappa$. The left panel corresponds to $\kappa = 1/640$, representing a weakly advection-dominated case, while $\kappa = 1/3100$ corresponds to a strongly advection-dominated case. The figure on the right shows the spectrum for a spatially varying $\kappa$, illustrating regions that are weakly or strongly advection-dominated depending on the location.
• Figure 2: The numerical 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 strongly advection-dominated scenarios.
• Figure 3: The numerical results for the weakly advection-dominated problem \ref{['problem1']} with $\kappa = \frac{1}{640}$. Figure (a) shows the total memory operations of MVMs as a function of the $L^2$ error, while Figure (b) depicts the achieved accuracy of the considered methods as a function of wall-clock time.
• Figure 4: The numerical results for the weakly advection-dominated problem \ref{['problem1']} with $\kappa = \frac{1}{640}$ are shown in the top two figures of Figure \ref{['fig1']}. These figures display the accuracy achieved and the computational cost for different methods evaluating $\exp(- (A_h + B_h)) u_0$ as a function of the time step size. The tolerances are set so that the exponential integrators reach accuracies of $10^{-4}$ (dashed-dotted line) and $10^{-7}$ (solid line), respectively. The bottom two panels show the accuracy of the methods as a function of computational cost for two selected time step sizes $\tau$.
• Figure 5: The top two panels present the numerical results for the strongly advection-dominated problem \ref{['problem1']} with $\kappa = \frac{1}{3100}$. These figures show 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 tolerances are set so that the exponential integrators reach target accuracies of $10^{-4}$ (dashed-dotted line) and $10^{-7}$ (solid line), respectively. The bottom two panels illustrate how the accuracy of these methods varies with computational cost for two selected time step values $\tau$.