Numerical analysis Ordinary differential equations Proposition

Exponential integrator

Builds on Initial value problem Matrix exponential Stiff equation

Statement

Let $A$ be a real $d \times d$ matrix, let $N \colon \R^d \to \R^d$ be continuous, and consider the initial value problem $y' = A y + N(y)$ , $y(t_0) = y_0$ . Put $\varphi_1(z) = (e^z - 1)/z$ for $z \ne 0$ and $\varphi_1(0) = 1$ , so that $\varphi_1(hA) = \sum_{k \ge 0} \frac{(hA)^k}{(k+1)!}$ .

The exact solution satisfies the variation-of-constants formula $y(t_0 + h) = e^{hA} y_0 + \int_0^h e^{(h - s)A} N(y(t_0 + s))\, ds,$ where $e^{hA}$ is the matrix exponential.
The exponential Euler method is the step $y_{n+1} = e^{hA} y_n + h\, \varphi_1(hA)\, N(y_n).$ If $N$ is constant, it gives the exact solution at every step, for every $h > 0$ .

Remarks

Clause 1 follows by differentiating $e^{-tA} y(t)$ . For clause 2, if $N \equiv b$ , the integral in clause 1 equals $\int_0^h e^{(h-s)A} b\, ds = h \varphi_1(hA) b$ , which is the method. In general, the exponential Euler method replaces $N(y(t_0 + s))$ by $N(y_n)$ in the integral; it has order $1$ when $N$ is Lipschitz and the solution is smooth. Example: for the scalar equation $y' = -100 y + 1$ the method reproduces $y(t) = 1/100 + (y_0 - 1/100) e^{-100 t}$ exactly, whatever the step size. Higher-order exponential Runge–Kutta and Rosenbrock-type methods use the functions $\varphi_k(z) = \sum_{j \ge 0} z^j / (j + k)!$ .

Going deeper

For a student

Exponential integrators are a tool for stiff equations where the stiffness sits in a known linear operator, such as the discretised Laplacian of a semilinear parabolic equation. They are explicit in $N$ and do not need the solution of nonlinear equations. They need products of $\varphi_k(hA)$ with vectors, computed with Krylov methods, Padé approximants or contour integrals. Their error analysis for parabolic problems uses that the bounds do not depend on the norm of $A$ .

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Tag	`H0MZB0DPP2`
URI	`m2://H0MZB0DPP2`
Kind	statement, proposition
Writing	W0 stub
MSC	65L05, 65F60, 65L04
Status	This copy of the site has no build data. Run `m2 status H0MZB0DPP2`.
About	Matrix exponential
Uses	Initial value problem, Matrix exponential, Stiff equation