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 the initial value problem described later in this article. 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 1960s, these methods were recognized by Certaine^[1] and Pope.^[2] As of late exponential integrators have become an active area of research, see Hochbruck and Ostermann (2010).^[3] Originally developed for solving stiff differential equations, the methods have been used to solve partial differential equations including hyperbolic as well as parabolic problems^[4] such as the heat equation.

Introduction

We consider initial value problems of the form,

${\displaystyle u'(t)=Lu(t)+N(u(t)),\qquad u(t_{0})=u_{0},\qquad \qquad (1)}$

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

${\displaystyle u'(t)=f(u(t)),\qquad u(t_{0})=u_{0},}$

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

${\displaystyle L={\frac {\partial f}{\partial u}}(u^{*});\qquad N=f(u)-Lu.}$

Here, ${\displaystyle {\frac {\partial f}{\partial u}}}$ refers to the partial derivative of $f$ with respect to $u$ (the Jacobian of f).

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:^[3]

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

This is similar to the exact integral used in the Picard–Lindelöf theorem. In the case of $N\equiv 0$, 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,^[5][6] linear multistep methods or a variety of other options.

Exponential Rosenbrock methods

Exponential Rosenbrock methods were shown to be very efficient in solving large systems of stiff ordinary differential equations, usually resulted from spatial discretization of time dependent (parabolic) PDEs. These integrators are constructed based on a continuous linearization of (1) along the numerical solution $u_{n}$

${\displaystyle u'(t)=L_{n}u(t)+N_{n}(u(t)),\qquad u(t_{0})=u_{0},\qquad (3)}$

where ${\displaystyle L_{n}={\frac {\partial f}{\partial u}}(u_{n}),N_{n}(u)=f(u)-L_{n}u.}$ This procedure enjoys the advantage, in each step, that ${\displaystyle {\frac {\partial N_{n}}{\partial u}}(u_{n})=0.}$ This considerably simplifies the derivation of the order conditions and improves the stability when integrating the nonlinearity ${\displaystyle N(u(t))}$. Again, applying the variation-of-constants formula (2) gives the exact solution at time $t_{n+1}$ as

${\displaystyle u(t_{n+1})=e^{h_{n}L_{n}}u(t_{n})+\int _{0}^{h_{n}}e^{(h_{n}-\tau )L_{n}}N_{n}(u(t_{n}+\tau ))d\tau .\qquad (4)}$

The idea now is to approximate the integral in (4) by some quadrature rule with nodes $c_{i}$ and weights ${\displaystyle b_{i}(h_{n}L_{n})}$ (${\displaystyle 1\leq i\leq s}$). This yields the following class of ${\displaystyle s-stage}$ explicit exponential Rosenbrock methods, see Hochbruck and Ostermann (2006), Hochbruck, Ostermann and Schweitzer (2009):

${\displaystyle U_{ni}=e^{c_{i}h_{n}L_{n}}u_{n}+h_{n}\sum _{j=1}^{i-1}a_{ij}(h_{n}L_{n})N_{n}(U_{nj}),}$

${\displaystyle u_{n+1}=e^{h_{n}L_{n}}u_{n}+h_{n}\sum _{i=1}^{s}b_{i}(h_{n}L_{n})N_{n}(U_{ni})}$

with ${\displaystyle u_{n}\approx u(t_{n}),U_{ni}\approx u(t_{n}+c_{i}h_{n}),h_{n}=t_{n+1}-t_{n}}$. The coefficients ${\displaystyle a_{ij}(z),b_{i}(z)}$ are usually chosen as linear combinations of the entire functions ${\displaystyle \varphi _{k}(c_{i}z),\varphi _{k}(z)}$, respectively, where

${\displaystyle \varphi _{0}(z)=e^{z},\quad \varphi _{k}(z)=\int _{0}^{1}e^{(1-\theta )z}{\frac {\theta ^{k-1}}{(k-1)!}}d\theta ,\quad k\geq 1.}$

These functions satisfy the recursion relation

${\displaystyle \varphi _{k+1}(z)={\frac {\varphi _{k}(z)-\varphi _{k}(0)}{z}},\ k\geq 0.}$

By introducing the difference ${\displaystyle D_{ni}=N_{n}(U_{ni})-N_{n}(u_{n})}$, they can be reformulated in a more efficient way for implementation (see also ^[3]) as

${\displaystyle U_{ni}=u_{n}+c_{i}h_{n}\varphi _{1}(c_{i}h_{n}L_{n})f(u_{n})+h_{n}\sum _{j=2}^{i-1}a_{ij}(h_{n}L_{n})D_{nj},}$

${\displaystyle u_{n+1}=u_{n}+h_{n}\varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}\sum _{i=2}^{s}b_{i}(h_{n}L_{n})D_{ni}.}$

In order to implement this scheme with adaptive step size, one can consider, for the purpose of local error estimation, the following embedded methods

${\displaystyle {\bar {u}}_{n+1}=u_{n}+h_{n}\varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}\sum _{i=2}^{s}{\bar {b}}_{i}(h_{n}L_{n})D_{ni},}$

which use the same stages ${\displaystyle U_{ni}}$ but with weights ${\displaystyle {\bar {b}}_{i}}$.

For convenience, the coefficients of the explicit exponential Rosenbrock methods together with their embedded methods can be represented by using the so-called reduced Butcher tableau as follows:

$c_{2}$
$c_{3}$	$a_{32}$
$\vdots$	$\vdots$		$\ddots$
$c_{s}$	$a_{s2}$	${\displaystyle a_{s3}}$	$\cdots$	$a_{s,s-1}$
	$b_{2}$	${\displaystyle b_{3}}$	$\cdots$	$b_{s-1}$	$b_{s}$
	${\displaystyle {\bar {b}}_{2}}$	${\displaystyle {\bar {b}}_{3}}$	$\cdots$	${\displaystyle {\bar {b}}_{s-1}}$	${\displaystyle {\bar {b}}_{s}}$

Stiff order conditions

Moreover, it is shown in Luan and Osterman (2014a) ^[7] that the reformulation approach offers a new and simple way to analyze the local errors and thus to derive the stiff order conditions for exponential Rosenbrock methods up to order 5. With the help of this new technique together with an extension of the B-series concept, a theory for deriving the stiff order conditions for exponential Rosenbrock integrators of arbitrary order has been finally given in Luan and Osterman (2013).^[8] As an example, in that work the stiff order conditions for exponential Rosenbrock methods up to order 6 have been derived, which are stated in the following table:

${\displaystyle {\begin{array}{|c|c|c| }\hline No.&{\text{Stiff order condition}}&{\text{Order}}\\\hline 1&\sum _{i=2}^{s}b_{i}(Z)c_{i}^{2}=2\varphi _{3}(Z)&3\\\hline 2&\sum _{i=2}^{s}b_{i}(Z)c_{i}^{3}=6\varphi _{4}(Z)&4\\\hline 3&\sum _{i=2}^{s}b_{i}(Z)c_{i}^{4}=24\varphi _{5}(Z)&5\\4&\sum _{i=2}^{s}b_{i}(Z)c_{i}K\psi _{3,i}(Z)=0&5\\\hline 5&\sum _{i=2}^{s}b_{i}(Z)c_{i}^{5}=120\varphi _{6}(Z)&6\\6&\sum _{i=2}^{s}b_{i}(Z)c_{i}^{2}M\psi _{3,i}(Z)=0&6\\7&\sum _{i=2}^{s}b_{i}(Z)c_{i}K\psi _{4,i}(Z)=0&6\\\hline \end{array}}}$

Here ${\displaystyle Z,K,M.\ }$ denote arbitrary square matrices.

Convergence analysis

The stability and convergence results for exponential Rosenbrock methods are proved in the framework of strongly continuous semigroups in some Banach space. For more details, see Hochbruck, Ostermann, and Schweitzer (2009), Luan and Ostermann (2014a, 2013).

Examples

All the schemes presented below fulfill the stiff order conditions and thus are also suitable for solving stiff problems.

Second-order method

The simplest exponential Rosenbrock method is the exponential Rosenbrock-Euler scheme, which has order 2, see, for example Hocbruck et al (2009):

${\displaystyle u_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n}).}$

Third-order methods

A class of third-order exponential Rosenbrock methods was derived in Hocbruck et al. (2009), named as exprb32, is given as:

exprb32:

1
	${\displaystyle 2\varphi _{3}}$
	0

which reads as

${\displaystyle U_{n2}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n}),}$

${\displaystyle u_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}\ 2\varphi _{3}(h_{n}L_{n})D_{n2},}$

where ${\displaystyle D_{n2}=N_{n}(U_{n2})-N_{n}(u_{n}).}$

For a variable step size implementation of this scheme, one can embed it with the exponential Rosenbrock-Euler:

${\displaystyle {\hat {u}}_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n}).}$

Fourth-order methods

According to Luan and Osterman (2016),^[9] the following fourth-order scheme, named as pexprb43, which can be not only implemented in serial but also in parallel computers, performs the best among the existing fourth-order exponential Rosenbrock methods:

pexprb43:

1/2
1
	${\displaystyle 16\varphi _{3}-48\varphi _{4}}$	${\displaystyle -2\varphi _{3}+12\varphi _{4}}$
	${\displaystyle 16\varphi _{3}}$	${\displaystyle -2\varphi _{3}}$

which reads as

${\displaystyle U_{n2}=u_{n}+{\frac {1}{2}}h_{n}\ \varphi _{1}({\frac {1}{2}}h_{n}L_{n})f(u_{n}),}$

${\displaystyle U_{n3}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n}),}$

${\displaystyle u_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}(16\varphi _{3}(h_{n}L_{n})-48\varphi _{4}(h_{n}L_{n}))D_{n2}+h_{n}(-2\varphi _{3}(h_{n}L_{n})+12\varphi _{4}(h_{n}L_{n}))D_{n3},}$

where ${\displaystyle D_{n2}=N_{n}(U_{n2})-N_{n}(u_{n}),D_{n3}=N_{n}(U_{n3})-N_{n}(u_{n}).\ }$

For a variable step size implementation of this scheme, one can use it together with the following third-order error estimator that uses the same internal stages:${\displaystyle U_{n2},U_{n3}}$:

${\displaystyle {\hat {u}}_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}16\varphi _{3}(h_{n}L_{n})D_{n2}+h_{n}(-2\varphi _{3}(h_{n}L_{n}))D_{n3}.}$

Fifth-order methods

A 3-stage fifth-order scheme

A 3-stage fifth-order exponential Rosenbrock scheme, called exprb53s3, was derived in Luan and Osterman (2014a), is given as:

exprb53s3:

1/2
9/10	${\displaystyle {\frac {27}{25}}\varphi _{3}({\frac {1}{2}}.)+{\frac {729}{125}}\varphi _{3}({\frac {9}{10}}.)}$
	${\displaystyle 18\varphi _{3}-60\varphi _{4}}$	${\displaystyle -{\frac {250}{81}}\varphi _{3}+{\frac {500}{27}}\varphi _{4}}$
	${\displaystyle 2\varphi _{3}}$	${\displaystyle {\frac {150}{81}}\varphi _{3}}$

which reads as

${\displaystyle U_{n2}=u_{n}+{\frac {1}{2}}h_{n}\ \varphi _{1}({\frac {1}{2}}h_{n}L_{n})f(u_{n}),}$

${\displaystyle U_{n3}=u_{n}+{\frac {9}{10}}h_{n}\ \varphi _{1}({\frac {9}{10}}h_{n}L_{n})f(u_{n})+{\big (}{\frac {27}{25}}\varphi _{3}({\frac {1}{2}}h_{n}L_{n})+{\frac {729}{125}}\varphi _{3}({\frac {9}{10}}h_{n}L_{n}){\big )}D_{n2},}$

${\displaystyle u_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}(18\varphi _{3}(h_{n}L_{n})-60\varphi _{4}(h_{n}L_{n}))D_{n2}+h_{n}(-{\frac {250}{81}}\varphi _{3}(h_{n}L_{n})+{\frac {500}{27}}\varphi _{4}(h_{n}L_{n}))D_{n3},}$

where ${\displaystyle D_{n2}=N_{n}(U_{n2})-N_{n}(u_{n}),D_{n3}=N_{n}(U_{n3})-N_{n}(u_{n}).\ }$

For a time-stepping version of this scheme, one can use it together with the following third-order error estimator (as it is not possible to embed it with a fourth-order scheme that uses the same internal stages ${\displaystyle U_{n2},U_{n3}}$):

${\displaystyle {\hat {u}}_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}2\varphi _{3}(h_{n}L_{n})D_{n2}+h_{n}({\frac {150}{81}}\varphi _{3}(h_{n}L_{n}))D_{n3}.}$

Parallel stages schemes

Similarly to the parallel exponential scheme pexprb43 given above, the following fifth-order (parallel stages) exponential Rosenbrock schemes can be also implemented both in serial and parallel computers, see Luan and Osterman (2016):

pexprb54s4: (4-stage)

1/4
1/2	${\displaystyle 4\varphi _{3}({\frac {1}{2}}.)}$
9/10	${\displaystyle {\frac {2916}{125}}\varphi _{3}({\frac {9}{10}}.)}$
	0	${\displaystyle 18\varphi _{3}-60\varphi _{4}}$	${\displaystyle -{\frac {250}{81}}\varphi _{3}+{\frac {500}{27}}\varphi _{4}}$
	${\displaystyle 64\varphi _{3}-60\varphi _{4}}$	${\displaystyle -8\varphi _{3}-{\frac {285}{8}}\varphi _{4}}$	${\displaystyle {\frac {125}{8}}\varphi _{4}}$

which reads as

${\displaystyle U_{n2}=u_{n}+{\frac {1}{4}}h_{n}\ \varphi _{1}({\frac {1}{4}}h_{n}L_{n})f(u_{n}),}$

${\displaystyle U_{n3}=u_{n}+{\frac {1}{2}}h_{n}\ \varphi _{1}({\frac {1}{2}}h_{n}L_{n})f(u_{n})+4\varphi _{3}({\frac {1}{2}}h_{n}L_{n})D_{n2},}$

${\displaystyle U_{n4}=u_{n}+{\frac {9}{10}}h_{n}\ \varphi _{1}({\frac {9}{10}}h_{n}L_{n})f(u_{n})+{\frac {2916}{125}}\varphi _{3}({\frac {9}{10}}h_{n}L_{n})D_{n2},}$

${\displaystyle u_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}(18\varphi _{3}(h_{n}L_{n})-60\varphi _{4}(h_{n}L_{n}))D_{n3}+h_{n}(-{\frac {250}{81}}\varphi _{3}(h_{n}L_{n})+{\frac {500}{27}}\varphi _{4}(h_{n}L_{n}))D_{n4},}$

where ${\displaystyle D_{n2}=N_{n}(U_{n2})-N_{n}(u_{n}),D_{n3}=N_{n}(U_{n3})-N_{n}(u_{n}),D_{n4}=N_{n}(U_{n4})-N_{n}(u_{n}).\ }$

For a time-stepping version of this scheme, one can use it together with the following fourth-order error estimator:

${\displaystyle {\hat {u}}_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}(64\varphi _{3}(h_{n}L_{n})-60\varphi _{4}(h_{n}L_{n}))D_{n2}+h_{n}(-8\varphi _{3}(h_{n}L_{n})-{\frac {285}{8}}\varphi _{4}(h_{n}L_{n}))D_{n3}+h_{n}{\frac {125}{8}}\varphi _{4}(h_{n}L_{n}))D_{n4}.}$

pexprb54s5: (5-stage)

1/2
1/2	${\displaystyle \varphi _{3}({\frac {1}{2}}.)}$
1/3	${\displaystyle {\frac {8}{27}}\varphi _{3}({\frac {1}{3}}.)}$
1	${\displaystyle 8\varphi _{3}}$
	0	${\displaystyle -32\varphi _{3}+384\varphi _{4}-1152\varphi _{5}}$	${\displaystyle 81\varphi _{3}-729\varphi _{4}}$	${\displaystyle \varphi _{3}-15\varphi _{4}}$
	0	${\displaystyle -32\varphi _{3}+384\varphi _{4}}$	${\displaystyle 81\varphi _{3}-729\varphi _{4}+1944\varphi _{5}}$	${\displaystyle \varphi _{3}-15\varphi _{4}+72\varphi _{5}}$

which reads as

${\displaystyle U_{n2}=u_{n}+{\frac {1}{2}}h_{n}\ \varphi _{1}({\frac {1}{2}}h_{n}L_{n})f(u_{n}),}$

${\displaystyle U_{n3}=u_{n}+{\frac {1}{2}}h_{n}\ \varphi _{1}({\frac {1}{2}}h_{n}L_{n})f(u_{n})+\varphi _{3}({\frac {1}{2}}h_{n}L_{n})D_{n2},}$

${\displaystyle U_{n4}=u_{n}+{\frac {1}{3}}h_{n}\ \varphi _{1}({\frac {1}{3}}h_{n}L_{n})f(u_{n})+{\frac {8}{27}}\varphi _{3}({\frac {1}{3}}h_{n}L_{n})D_{n2},}$

${\displaystyle U_{n5}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+8\varphi _{3}(h_{n}L_{n})D_{n2},}$

${\displaystyle u_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}(-32\varphi _{3}(h_{n}L_{n})+384\varphi _{4}(h_{n}L_{n})-1152\varphi _{5}(h_{n}L_{n}))D_{n3}+h_{n}(81\varphi _{3}(h_{n}L_{n})-729\varphi _{4}(h_{n}L_{n})+1944\varphi _{5}(h_{n}L_{n}))D_{n4}+h_{n}(\varphi _{3}(h_{n}L_{n})-15\varphi _{4}(h_{n}L_{n})+72\varphi _{5}(h_{n}L_{n}))D_{n5},}$

where ${\displaystyle D_{n2}=N_{n}(U_{n2})-N_{n}(u_{n}),D_{n3}=N_{n}(U_{n3})-N_{n}(u_{n}),D_{n4}=N_{n}(U_{n4})-N_{n}(u_{n}),D_{n5}=N_{n}(U_{n5})-N_{n}(u_{n}).\ }$

For the purpose of time-stepping, one can use it together with the following fourth-order error estimator:

${\displaystyle {\hat {u}}_{n+1}=u_{n}+h_{n}\ \varphi _{1}(h_{n}L_{n})f(u_{n})+h_{n}(-32\varphi _{3}(h_{n}L_{n})+384\varphi _{4}(h_{n}L_{n}))D_{n3}+h_{n}(81\varphi _{3}(h_{n}L_{n})-729\varphi _{4}(h_{n}L_{n}))D_{n4}+h_{n}(\varphi _{3}(h_{n}L_{n})-15\varphi _{4}(h_{n}L_{n}))D_{n5}.}$

First-order forward Euler exponential integrator

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

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

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, see Hochbruck and Ostermann (2010), Tokman (2006, 2011). See references for further details of where these functions come from.^[5]

Fourth-order ETDRK4 method of Cox and Mathews

Cox and Mathews^[10] describe a fourth-order method exponential time differencing (ETD) method that they used Maple to derive.

We use their notation, and assume that the unknown function is $u$, and that we have a known solution $u_{n}$ at time $t_{n}$. Furthermore, we'll make explicit use of a possibly time dependent right hand side: ${\mathcal {N}}={\mathcal {N}}(u,t)$.

Three stage values are first constructed:

$a_{n}=e^{{Lh/2}}u_{n}+L^{{-1}}\left(e^{{Lh/2}}-I\right){\mathcal {N}}(u_{n},t_{n})$

$b_{n}=e^{{Lh/2}}u_{n}+L^{{-1}}\left(e^{{Lh/2}}-I\right){\mathcal {N}}(a_{n},t_{n}+h/2)$

$c_{n}=e^{{Lh/2}}a_{n}+L^{{-1}}\left(e^{{Lh/2}}-I\right)\left(2{\mathcal {N}}(b_{n},t_{n}+h/2)-{\mathcal {N}}(u_{n},t_{n})\right)$

The final update is given by,

$u_{{n+1}}=e^{{Lh}}u_{n}+h^{{-2}}L^{{-3}}\left\{\left[-4-Lh+e^{{Lh}}\left(4-3Lh+(Lh)^{2}\right)\right]{\mathcal {N}}(u_{n},t_{n})+2\left[2+Lh+e^{{Lh}}\left(-2+Lh\right)\right]\left({\mathcal {N}}(a_{n},t_{n}+h/2)+{\mathcal {N}}(b_{n},t_{n}+h/2)\right)+\left[-4-3Lh-(Lh)^{2}+e^{{Lh}}\left(4-Lh\right)\right]{\mathcal {N}}(c_{n},t_{n}+h)\right\}.$

If implemented naively, the above algorithm suffers from numerical instabilities due to floating point round-off errors.^[11] To see why, consider the first function,

$\varphi _{1}(z)={\frac {1-e^{z}}{z}},$

which is present in the first-order Euler method, as well as all three stages of ETDRK4. For small values of $z$, this function suffers from numerical cancellation errors. However, these numerical issues can be avoided by evaluating the $\varphi _{1}$ function via a contour integral approach ^[11] or by a Padé approximant.^[12]

Applications

Exponential integrators are used for the simulation of stiff scenarios in scientific and visual computing, for example in molecular dynamics,^[13] for VLSI circuit simulation,^[14] and in computer graphics.^[15] They are also applied in the context of hybrid monte carlo methods.^[16] In these applications, exponential integrators show the advantage of large time stepping capability and high accuracy. To accelerate the evaluation of matrix functions in such complex scenarios, exponential integrators are often combined with Krylov subspace projection methods.

See also

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

Notes

1. Certaine (1960)
2. Pope (1963)
3. Hochbruck and Ostermann, (2010)
4. Hochbruck and Ostermann, (2006)
5. Cox and Mathews (2002)
6. Tokman (2006, 2011)
7. Luan and Osterman (2014a)
8. Luan and Osterman (2013)
9. Luan and Osterman (2016)
10. Cox and Mathews 2002
11. Kassam and Trefethen (2005)
12. Berland et al. (2007)
13. Michels and Desbrun (2015)
14. Zhuang et al. (2014)
15. Michels et al. (2014)
16. Chao et al. (2015)

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. doi:10.1137/040612683.
• Berland, Havard; Skaflestad, Bard; Wright, Will M. (2007). "EXPINT-A MATLAB Package for Exponential Integrators". ACM Transactions on Mathematical Software. 33 (1). doi:10.1145/1206040.1206044.
• Chao, Wei-Lun; Solomon, Justin; Michels, Dominik L.; Sha, Fei (2015). "Exponential Integration for Hamiltonian Monte Carlo". Proceedings of the 32nd International Conference on Machine Learning (ICML-15): 1142–1151.
• Certaine, John (1960). The solution of ordinary differential equations with large time constants. Wiley. pp. 128–132.
• Cox, S. M.; Mathews, P.C. (March 2002). "Exponential time differencing for stiff systems". Journal of Computational Physics. 176 (2): 430–455. doi:10.1006/jcph.2002.6995.
• Hochbruck, Marlis; Ostermann, Alexander (May 2010). "Exponential integrators". Acta Numerica. 19: 209–286. doi:10.1017/S0962492910000048.
• 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 (May 2005). "Exponential Runge–Kutta methods for parabolic problems". Applied Numerical Mathematics. 53 (2-4): 323–339. doi:10.1016/j.apnum.2004.08.005.
• Luan, Vu Thai; Ostermann, Alexander (2014a). "Exponential Rosenbrock methods of order five-construction, analysis and numerical comparisons". Journal of Computational and Applied Mathematics. 255: 417–431. doi:10.1016/j.cam.2013.04.041.
• Luan, Vu Thai; Ostermann, Alexander (2014c). "Explicit exponential Runge-Kutta methods of high order for parabolic problems". Journal of Computational and Applied Mathematics. 256: 168–179. doi:10.1016/j.cam.2013.07.027.
• Luan, Vu Thai; Ostermann, Alexander (2013). "Exponential B-series: The stiff case". SIAM Journal of Numerical Analysis. 51: 3431–3445. doi:10.1137/130920204.
• Luan, Vu Thai; Ostermann, Alexander (2014b). Stiff order conditions for exponential Runge-Kutta methods of order five. Modeling, Simulation and Optimization of Complex Processes - HPSC 2012 (H.G. Bock et al. eds.). pp. 133–143. doi:10.1007/978-3-319-09063-4_11.
• Luan, Vu Thai; Ostermann, Alexander (2016). "Parallel exponential Rosenbrock methods". Computers and mathematics with Applications. doi:10.1016/j.camwa.2016.01.020.
• Michels, Dominik L.; Desbrun, Mathieu (2015). "A Semi-analytical Approach to Molecular Dynamics". Journal of Computational Physics. 303: 336–354. doi:10.1016/j.jcp.2015.10.009.
• Michels, Dominik L.; Sobottka, Gerrit A.; Weber, Andreas G. (2014). "Exponential Integrators for Stiff Elastodynamic Problems". ACM Transactions on Graphics. 33: 7:1–7:20. doi:10.1145/2508462.
• Pope, David A (1963). "An exponential method of numerical integration of ordinary differential equations". Communications of the ACM. 6 (8): 491–493. doi:10.1145/366707.367592.
• Tokman, Mayya (October 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.
• Tokman, Mayya (April 2006). "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.
• Trefethen, Lloyd N.; Aly-Khan Kassam (2005). "Fourth-Order Time-Stepping for Stiff PDEs". SIAM Journal of Scientific Computing. 26 (4): 1214–1233. doi:10.1137/S1064827502410633.
• Zhuang, Hao; Weng, Shih-Hung; Lin, Jeng-Hau; Cheng, Chung-Kuan (2014). MATEX: A Distributed Framework for Transient Simulation of Power Distribution Networks. (PDF). ACM/IEEE Proceedings of The 51st Annual Design Automation Conference (DAC). doi:10.1145/2593069.2593160.

External links

• integrators on GPGPUs
• code for a meshfree exponential integrator