## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2011, Special Issue (2011), Article ID 193781, 27 pages.

### Computing Exponential for Iterative Splitting Methods: Algorithms and Applications

Jürgen Geiser

#### Abstract

Iterative splitting methods have a huge amount to compute matrix exponential. Here, the acceleration and recovering of higher-order schemes can be achieved. From a theoretical point of view, iterative splitting methods are at least alternating Picards fix-point iteration schemes. For practical applications, it is important to compute very fast matrix exponentials. In this paper, we concentrate on developing fast algorithms to solve the iterative splitting scheme. First, we reformulate the iterative splitting scheme into an integral notation of matrix exponential. In this notation, we consider fast approximation schemes to the integral formulations, also known as $\varphi$-functions. Second, the error analysis is explained and applied to the integral formulations. The novelty is to compute cheaply the decoupled exp-matrices and apply only cheap matrix-vector multiplications for the higher-order terms. In general, we discuss an elegant way of embedding recently survey on methods for computing matrix exponential with respect to iterative splitting schemes. We present numerical benchmark examples, that compared standard splitting schemes with the higher-order iterative schemes. A real-life application in contaminant transport as a two phase model is discussed and the fast computations of the operator splitting method is explained.

#### Article information

Source
J. Appl. Math., Volume 2011, Special Issue (2011), Article ID 193781, 27 pages.

Dates
First available in Project Euclid: 29 August 2011

https://projecteuclid.org/euclid.jam/1314650240

Digital Object Identifier
doi:10.1155/2011/193781

Mathematical Reviews number (MathSciNet)
MR2784394

Zentralblatt MATH identifier
1217.65065

#### Citation

Geiser, Jürgen. Computing Exponential for Iterative Splitting Methods: Algorithms and Applications. J. Appl. Math. 2011, Special Issue (2011), Article ID 193781, 27 pages. doi:10.1155/2011/193781. https://projecteuclid.org/euclid.jam/1314650240