Exponential integrator

Exponential integrators are a class of numerical methods for the solution of partial and ordinary differential equations. This large class of methods from numerical analysis is based on the exact integration of the linear part of a problem. Because the linear part is integrated exactly, this can help to mitigate the stiffness of a differential equation. Exponential integrators can be constructed to be explicit or implicit for numerical ordinary differential equations or serve as the time integrator for numerical partial differential equations.

Background

Dating back to at least the 1960's, these methods were recognized by Certain and Pope. As of late exponential integrators have become an active area of research. Originally developed for solving stiff differential equations, the methods have been used to solve partial differential equations including hyperbolic as well as parabolic problems such as the heat equation.

Introduction

We consider initial value problems of the form,

$y'(t)=Ly(t)+{\mathcal {N}}(y(t)),\qquad y(t_{0})=y_{0},\qquad \qquad (1)$

where $L$ is composed of linear terms, and ${\mathcal {N}}$ is composed of the non-linear terms. These problems can come from a more typical initial value problem

$y'(t)=f(y(t)),\qquad y(t_{0})=y_{0},$

after linearizing locally about a fixed or local state $y^{*}$:

$L={\frac {\partial f}{\partial y}}(y^{*});\qquad {\mathcal {N}}=f(y)-Ly.$

Here, ${\frac {\partial f}{\partial y}}$ refers to the partial derivative of $f$ with respect to $y$.

Exact integration of this problem from time 0 to a later time $t$ can be performed using matrix exponentials to define an integral equation for the exact solution:

$y(t)=e^{Lt}y_{0}+\int _{0}^{t}e^{L(t-\tau )}{\mathcal {N}}\left(y\left(\tau \right)\right)\,d\tau .\qquad (2)$

This is similar to the exact integral used in the Picard–Lindelöf theorem. In the case of ${\mathcal {N}}\equiv 0$, this formulation is the exact solution to the linear differential equation.

Numerical methods require a discretization of equation (2). They can be based on Runge-Kutta discretizations , Taylor series.

Examples

See also: the first-order exponential integrator for more details.

The simplest method is based on a forward Euler time discretization. It can be realized by holding the term ${\mathcal {N}}(y(\tau ))\approx {\mathcal {N}}(y(0))$ constant over the whole interval. Exact integration of $e^{L(t-\tau )}$ then results in the

$y(t)=e^{Lt}y_{0}+L^{-1}(1-e^{Lt}){\mathcal {N}}(y(t_{0})).$

Of course, this process can be repeated over small intervals to serve as the basis of a single-step numerical method.

In general, one defines a sequence of functions,

$\varphi _{0}(z)=e^{z};\qquad \varphi _{1}(z)={\frac {e^{z}-1}{z}}\qquad \varphi _{2}(z)={\frac {e^{z}-1-z}{z^{2}}}$

that show up in these methods. Usually, these linear operators are not computed exactly, but a Krylov subspace iterative method can be used to efficiently compute the multiplication of these operators times vectors efficiently. See references for further details of where these functions come from.,

See also

• General linear methods
• Linear multistep methods
• Numerical analysis
• Numerical methods for ordinary differential equations
• Runge-Kutta methods

References

• Berland, Havard; Owren, Brynjulf; Skaflestad, Bard (2005). "B-series and Order Conditions for Exponential Integrators". SIAM Journal of Numerical Analysis. 43 (4): 1715–1727.
• Certaine, John (1960). The solution of ordinary differential equations with large time constants. Wiley. pp. 128–132. {{cite book}}: Unknown parameter |booktitle= ignored (help)
• Cox, S. M.; Mathews, P.C. (2002). "Exponential time differencing for stiff systems". Journal of Computational Physics. 176 (2): 430–455. doi:10.1006/jcph.2002.6995. {{cite journal}}: Unknown parameter |month= ignored (help)
• Hochbruck, Marlis; Ostermann, Alexander (2010). "Exponential integrators". Acta Numer. 19: 209–286.
• Hochbruck, Marlis; Ostermann, Alexander (2005). "Explicit exponential Runge-Kutta methods for semilinear parabolic problems". SIAM Journal of Numerical Analysis. 43 (3): 1069–1090. doi:10.1137/040611434.
• Hochbruck, Marlis; Ostermann, Alexander (2005). "Exponential Runge–Kutta methods for parabolic problems". Applied Numerical Mathematics. 53 (2–4): 323–339. doi:10.1016/j.apnum.2004.08.005. {{cite journal}}: Unknown parameter |month= ignored (help)
• Koskela, Antti; Ostermann, Alexander (2013). "Exponential Taylor methods: Analysis and implementation". Computers & Mathematics with Applications. 65 (3): 487–499. doi:10.1016/j.camwa.2012.06.004. {{cite journal}}: Unknown parameter |month= ignored (help)
• Pope, David A (1963). "An exponential method of numerical integration of ordinary differential equations". Communications of the ACM. 6 (8): 491–493.
• Tokman, Mayya (2011). "A new class of exponential propagation iterative methods of Runge–Kutta type (EPIRK)". Journal of Computational Physics. 230 (24): 8762–8778. doi:10.1016/j.jcp.2011.08.023. {{cite journal}}: Unknown parameter |month= ignored (help)
• Tokman, Mayya. "Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods". Journal of Computational Physics. 213 (2): 748–776. doi:10.1016/j.jcp.2005.08.032. {{cite journal}}: Unknown parameter |month= ignored (help)
• Trefethen, Lloyd N. (2005). "Fourth-Order Time-Stepping for Stiff PDEs". SIAM Journal of Scientific Computing. 26 (4): 1214–1233. doi:10.1137/S1064827502410633. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• integrators on GPGPUs
• code for a meshfree exponential integrator