Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems.
This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value 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.
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).
can be performed using matrix exponentials to define an integral equation for the exact solution:[3] This is similar to the exact integral used in the Picard–Lindelöf theorem.
, 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][7] linear multistep methods or a variety of other options.
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
This considerably simplifies the derivation of the order conditions and improves the stability when integrating the nonlinearity
Again, applying the variation-of-constants formula (2) gives the exact solution at time
as The idea now is to approximate the integral in (4) by some quadrature rule with nodes
, respectively, where These functions satisfy the recursion relation By introducing the difference
, they can be reformulated in a more efficient way for implementation (see also [3]) as 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: Moreover, it is shown in Luan and Ostermann (2014a)[8] 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 Ostermann (2013).
The stability and convergence results for exponential Rosenbrock methods are proved in the framework of strongly continuous semigroups in some Banach space.
The simplest exponential Rosenbrock method is the exponential Rosenbrock–Euler scheme, which has order 2, see, for example Hochbruck et al. (2009): A class of third-order exponential Rosenbrock methods was derived in Hochbruck 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: Cox and Matthews[5] describe a fourth-order method exponential time differencing (ETD) method that they used Maple to derive.
Furthermore, we'll make explicit use of a possibly time dependent right hand side:
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.
[10] To see why, consider the first function, which is present in the first-order Euler method, as well as all three stages of ETDRK4.
function via a contour integral approach [10] or by a Padé approximant.
[11] Exponential integrators are used for the simulation of stiff scenarios in scientific and visual computing, for example in molecular dynamics,[12] for VLSI circuit simulation,[13][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.