الفهرس 
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Abstract
The objectives of this thesis are to design, analyze and numerically investigate easily implementable Exponential Peer Methods (EPMs) for ordinary differential equations (ODEs), where the problem splits into a linear stiff and a nonlinear non-stiff part. The spatial discretization of time-dependent partial differential equations (PDEs) in general leads to such systems.
Chapter 1
This chapter is devoted to give a brief introduction to the concept of stiff problems, the phenomenon of numerical stiffness is explained, and to exponential integrators as alternative numerical methods developed to overcome the phenomenon of stiffness. Mathematical background material that we need later in the thesis is collected. In particular, we introduce Lipschitz condition and the logarithmic norm. A MATLAB package called EXPINT is used as a tool for testing and comparison of exponential integrators for constant step sizes
Chapter 2
This chapter is devoted to give an overview about the derivation, analysis, implementation and evaluation of exponential peer methods for constant step sizes. Consistency and stability of the methods are investigated, and we formulate simplifying conditions which guarantee order p = s-1, where s is the number of stages. For the non-stiff case the order is p = s. A special class of EPMs of stiff order p = s-1 with only two different arguments for the exponential functions is studied, and by a special choice of the nodes we obtain optimally zero-stable methods. We show that the methods solve linear problems exactly. Numerical order tests show the theoretically obtained orders.
Chapter 3
This chapter is a generalization for Chapter 3, for variable step sizes, and the idea of methods with an adaptive step size control is described. order conditions for the coefficients, which now will depend on the step size ratio are derived. We present one subclass which is optimally zero stable for all step size sequences. For another special class of methods with only two different arguments in the functions we prove stiff order p = s-1. For this class we compute bounds on the step size ratio which guarantee zero-stability in the non-stiff case. These bounds are fairly large for practical computations. In the stiff case we show convergence of stiff order p = s- 1 under mild restrictions of the step size sequence.
Chapter 4 and 5
Numerical results obtained using the framework of the EXPINT package for the new methods are reported and analyzed in this chapter for constant and variable step sizes. For special problem types the exponential peer methods turn out to be comparable and superior, but for others the classical codes are more efficient. The most expensive part in EPMs is the computation of the functions. Better numerical methods for this task will highly improve the performance of the methods.