A family of high-order accurate contour integral methods for strongly continuous semigroups

Abstract

Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time approximations that can be efficiently evaluated anywhere within a finite time horizon. However, there are core theoretical challenges that restrict their use cases to analytic semigroups, e.g., parabolic equations. In this article, we use carefully regularized contour integral representations to construct a family of new high-order quadrature schemes for the larger, less regular, class of strongly continuous semigroups. Our algorithms are accompanied by explicit high-order error bounds and near-optimal parameter selection. We demonstrate key features of the schemes on singular first-order PDEs from Koopman operator theory.

Figures (10)

• Figure 1: When $A$ generates an analytic semigroup, its spectrum is contained in a sector (shaded grey) and efficient contour integral methods leverage a rapidly decaying integrand along the contour (blue line) in the left half-plane colbrook2022computing. For strongly continuous semigroups, the spectrum may fill the whole left half-plane and standard contour integral representations may have a slowly decaying integrand.
• Figure 2: In the left-hand panel, the total error bound (for $E_D+E_T$) from \ref{['thm:trap_rule_1_error']} is plotted as a function of $N=$ 'number of quadrature nodes' at $t=1$ with $\delta=a=2$ when the discretization parameter is fixed at $h=0.5$ (blue circles) and optimized numerically to minimize the total error bound (orange circles). The bounds are plotted with the normalization $\|(\delta+a-A)^2x\|=1$ and compared with the asymptotic convergence rate $\mathcal{O}(1/N)$ (black dashed line). The right panel illustrates the decay of the integrand along the contour due to the regularizer $r(z)=(\delta+a-z)^{-2}$.
• Figure 3: The total error bound (for $E_D+E_T$) from \ref{['thm:trap_rule_1_error']} is plotted as a function of $a=$ 'pole location' of the regularizar $r(z)=(\delta+a-z)^{-2}$ for contour locations $\delta = 3$ (left panel) and $\delta=5$ (left panel) and $N=100$ (blue), $N=200$ (red), $N=400$ (yellow), and $N=800$ (purple). In both panels, the error bound was computed at $t=1$ and normalized so that $\|(\delta+a-A)^2x\|=(\delta+a)^2$. The discretization parameter $h$ was optimized numerically to minimize the total error bound.
• Figure 4: In the left-hand panel, the total error bounds (for $E_D+E_T$) from \ref{['thm:trap_rule_1_error']} are plotted as a function of $N=$ 'number of quadrature nodes' at $t=1$ with $\delta=2$ for $m=2,4,6,8$. The discretization parameter is optimized numerically to minimize the total error bound. The bounds are plotted with the normalizations $\|(2\delta-A)^mx\|=2^{m-2}$ to reflect typical graph norm growth of smooth functions, and compared with the asymptotic convergence rates $\mathcal{O}(1/N^{m-1})$ (black dashed lines). The right-hand panel shows the integrand's decay on the contour due to the regularizer $r(z)=(2\delta-z)^{-m}$.
• Figure 5: The left panel compares the total error bounds (for $E_D+E_T$) from \ref{['thm:trap_rule_1_error']}, plotted as a function of $N=$ 'number of quadrature nodes' at $t=1$ with $\sigma=\delta=2$ for $m=2,4,6,8$. The discretization parameters $N$ and $h$ are chosen according to \ref{['eqn:choose_h', 'eqn:choose_n']} for target error tolerances of $\epsilon=10^{-1},10^{-2},\ldots,10^{-8}$ (dotted lines) and compared with the error bound for optimal $h$ at the same values of $N$ (solid lines). For $N\geq 10$, the two are nearly indistinguishable, indicating that \ref{['eqn:choose_h', 'eqn:choose_n']} are nearly optimal. All bounds are normalized with $\|(2\delta-A)^mx\|=2^{m-2}$. The right panel demonstrates exponential convergence for functions in $D(A^\infty)$ with norm $\|(2\delta-A)^mx\|\leq (2\delta)^{m-2}$, reflected in the linear envelope (black dashes) of the $m$th order schemes on the semilog plot.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Lemma 1
• proof
• proof : Proof of discretization error in \ref{['thm:trap_rule_2_error']}
• proof : Proof of truncation error in \ref{['thm:trap_rule_2_error']}