Effective Exponential Integral And Exponential Matrix Matrix Function And Vector Product

Exponential integrators are an effective class of numerical methods for the numerical solution of semi-linear problems. The main computational cost of exponential integrators is dominated by the need to compute the products of exponential-like matrix functions φ(A) and vector b.In this thesis, we consider to construct a flexible variant of exponential integration methods for large systems of differential equations involving terms of different types, that is Such problems typically arise from many scientific and engineering applications, including me-chanical, molecular dynamics, oceanography and so on. Typical examples include reaction-diffusion equations and Navier-Stokes equations. When the right-hand side of the differential equations (3) has terms with different characteristics, it is natural to consider methods that take advantage of the structure of F. We describe a flexible variant of exponential integration methods for large systems of differential equations (3). The version possesses the flexibility and generality which allows to further exploit the special structure of the system. By using modified B-series and bi-coloured rooted trees, we can derive the general structure of the classical order conditions for these schemes. Some numerical schemes are constructed and their order conditions are derived. A general error analysis is presented to show the converge results based on abstract framework of analytic semigroups. Numerical experiments with reaction-diffusion type problems are included.It is noticed that the evaluation of these φ-functions plays important roles in the implemen-tation of exponential integration methods, we also consider to construct effective numerical algo-rithms for solving the linear combination of φ-functions acting on certain vectors: In fact, the implementation of exponential integrators is virtually to compute a certain number of expressions of the form (4) several times at each time step. The effective evaluation of (4), which is crucial to the effectiveness of exponential integrators, has been extensively investigated in recent years. Overall, the other parts try to construct numerical algorithms for solving (4).Firstly, we develop numerical algorithms for computing expression of the form (4) acting on a large and sparse matrix A∈Rn×n. By reducing the original problem based on block Krylov subspaces methods, we derive a single low dimensional matrix exponential to approximate (4). We do not need to evaluate all the φ-functions, so the methods can be performed without too much effort. The whole process can be accomplished by appropriately projecting a corresponding systems of large ODEs onto a block Krylov subspace and then accurately solving the reducing systems of ODEs by a matrix exponential. We obtain an infinite series error expansion, which is direct analogue of Saad’s result on the classical Arnoldi/Lanczos approximation to eAb. By using the first term of error expansion, we derive two reliable a posteriori estimates and a corrected scheme with higher accuracy. To avoid the growth of memory requirements and computation cost, we embed the time-step methods into our algorithms to satisfy the accuracy requirements under a moderate dimension of the block Krylov subspace. Some typical numerical tests illustrate that our algorithm can achieve high accuracy, which is more efficient than some codes in the literature.Secondly, our interest is the case of ill conditioned matrices A with a large norm typical arising from the discretization of stiff time-dependent parabolic PDEs or sectorial operators. The widely used standard Krylov subspace methods often turn out to be inefficient for this type of matrices because of slow convergence and large memory requirement. To overcome these prob-lems, we describe how a block shift-and-invert Krylov subspace method can be employed for the linear combinations of the action of φ-functions of the form (4). By using exact error expansion, we provide some reliable a posteriori estimates and effective corrected schemes from a practical viewpoint. We present several numerical experiments to demonstrate error bounds derived and show the efficiency of our algorithm over two commonly used numerical algorithms. Numerical results exhibits that the proposed algorithm has faster convergence speed and the a posteriori error estimates can capture correctly the characteristics of the true error.