Research Of Parallel Algorithms For Solving Numerical Solutions Of Several Classes Partial Differential Equations

Solving the problem of partial differential equations (PDE) has been paid attention by people all the time. It is difficult to find the solution of PDE because it depends on not only the definite condition, but also the area of variable. So solving numerical solution of PDE becomes the focus of research now. The number of node should be increasing to reduce the error and it increases the calculation. There are lots of methods to reduce the calculation but many problems cannot be solved within the capacity of computers nowadays. Fortunately,the computers are popular and we can use multiple computers to solve the same problem.So the parallel algorithms exist and studying parallel algorithms of numerical solution of PDE also becomes an important research direction.In the paper, chapter 2 describes Foster algorithms which is composed of division,communication, gather and mapping and parallel realization which is able to solve some simple equations. Based on chapter 3, using foster algorithms, chapter 4 gives the realization of the four steps process and analyzes the communication costs. Finally, a numerical example is given to verify the correctness of the method, and we compare the experimental data and get reduction ratio.Usually, the majority of problems in PDE can be transformed into the problems of solving banded liner equations, according to dividing from linear equations by line in Foster algorithms, we can gain the original task. Under the distributed communication, the original task can be gathered by a mapping of one of tasks and processor, finally we can get a new matrix equations iterative algorithm whose computing coefficients are banded or similar banded. And a numerical example is given to verify the method. According to the process of algorithm design and the computer simulation, we can find that the algorithm has very strong parallelism. Division reduces the order of finding the inverse matrix. In the iteration,we only calculate it once. When error is small, the number of iterations is 22. We can find that error appears in the first few component of the solution from the simulation diagram and this algorithm has a higher calculating precision.