Parallel Rosenbrock Methods For Stiff Differential Equations

With the rapid development of parallel computer, the parallel computing has become one of the most important means in solving stiff differential equations, so it is urgent to study the efficient parallel algorithms of stiff differential equations. A lot of literature has studied the parallel computing of Runge-Kutta methods and block methods, but when it comes to the sequential Rosenbrock methods (ROWs), which have been proved to be performed efficiently in a sequential computing environment, few literature studies whether they can construct the relevant parallel computing formulae. In 1996, Cheng Lirong and Liu Degui first constructed a class of parallel Rosenbrock methods (PRMs) for solving stiff ordinary differential equations. The computational speed of these methods is higher than that of sequential Rosenbrock methods, but the computational accuracy of these methods is inferior to that of the latter. This paper is trying to study and construct the efficient parallel Rosenbrock methods, which are superior to corresponding sequential Rosenbrock methods in the aspects of computational speed and computational accuracy, and to make a thorough study on the problems of solving the stiff ordinary differential equations, the differential-algebraic equations, the stiff delay differential equations, the initial boundary value problems for partial differential equations and the real-time digital simulations of stiff dynamic systems by using the new parallel algorithms. In Chapter 2, by making the improvement of PRMs, a class of modified paral- lel Rosenbrock methods (MPROWs) is constructed. Convergence and stability of these methods are discussed. Especially, using the Powell method to optimize the stability re- gions of the method and minimize the error constant of the method, we search out the practically optimal values of the free parameters, and get correspondingly MPROW3 of two-stage third-order and MPROW4 of three-stage fourth-order, which are all A-stable. Theoretical analysis and numerical experiments show the new methods not only keep the advantage of rapid speed of PRMs, but also obviously improve the numerical stability and computational accuracy of PRMs. The new methods are more efficient than the existing parallel and sequential Rosenbrock methods for solving stiff problems. In Chapter 3, by making further extension of MPROWs, a class of parallel extended Rosenbrock methods (PEROWs) is presented. Convergence and stability of PEROWs are discussed. Particularly, A-stable PEROW4 formula of two-stage fourth-order is obtained by choosing free parameters appropriately. The convergence order of the method has one more order than that of MPROW3 of the same stage. So it stands obviously in an advantage position in the computational accuracy. Moreover, the computation workload assigned to each processor is roughly balanced, so the PEROW4 of two-stage fourth-order can obtained very ideal speed-up and parallel efficiency. At present, the software package of stiff differential equations based on the MPROWs and PEROWs has been initially constructed. And using Hairer抯 and Wanner抯 tech- niques, it is also applicable to solve the differential-algebraic equations of index one. The numerical experiment of the differential-algebraic equations confirms that the constructed methods are efficient and reliable from the other aspect. It is an important tendency to introduce the idea of parallel processing into real- time digital simulation. However in some real-time digital simulation parallel algorithms III...