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.
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Background
Dating back to at least the 1960s, these methods were recognized by Certaine and Pope. As of late exponential integrators have become an active area of research, see Hochbruck and Ostermann (2010). 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,
where
after linearizing locally about a fixed or local state
Here,
Exact integration of this problem from time 0 to a later time
This is similar to the exact integral used in the Picard–Lindelöf theorem. In the case of
Numerical methods require a discretization of equation (2). They can be based on Runge-Kutta discretizations, 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
where
The idea now is to approximate the integral in (4) by some quadrature rule with nodes
with
These functions satisfy the recursion relation
By introducing the difference
In order to implement this scheme with adaptive step size, one can consider, for the purpose of local error estimation, the following embedded methods
which use the same stages
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:
Stiff order conditions
Moreover, it is shown in Luan and Osterman (2014a) 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). 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:
Here
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):
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:
which reads as
where
For a variable step size implementation of this scheme, one can embed it with the exponential Rosenbrock-Euler:
Fourth-order methods
According to Luan and Osterman (2016), 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:
which reads as
where
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:
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:
which reads as
where
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
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)
which reads as
where
For a time-stepping version of this scheme, one can use it together with the following fourth-order error estimator:
pexprb54s5: (5-stage)
which reads as
where
For the purpose of time-stepping, one can use it together with the following fourth-order error estimator:
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
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,
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.
Fourth-order ETDRK4 method of Cox and Mathews
Cox and Mathews 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
Three stage values are first constructed:
The final update is given by,
If implemented naively, the above algorithm suffers from numerical instabilities due to floating point round-off errors. To see why, consider the first function,
which is present in the first-order Euler method, as well as all three stages of ETDRK4. For small values of
Applications
Exponential integrators are used for the simulation of stiff scenarios in scientific and visual computing, for example in molecular dynamics, for VLSI circuit simulation, and in computer graphics. They are also applied in the context of hybrid monte carlo methods. 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.