Software Development For Numerical Solutions Of Partial Differential Equations With Finite Difference Method

This paper mainly develops a numerical solution software for the partial differential equations(PDEs)based on the Internet.The main function of the software is to solve the parabolic,hyperbolic and elliptic partial differential equations.This software can reduce time and effort of the researchers when they repeat programming in the process of numerical experiments or in-depth understanding of numerical methods,and improve the efficiency of scientific research by providing a convenient computing platform and learning platform for the learners and researchers in the field of numerical solutions of PDEs.The software integrates 240 finite difference schemes for 41 basic equations,including a few classic low precision schemes,and the newly developed high-precision compact schemes in recent thirty years(with fourth.order and even higher order accuracy).The software includes two grid computing models,i.e.,uniform and nonuniform grid computing,as well as two methods,i.e.,single layer grid iteration method and multigrid iteration method.The software provides numerical solutions,exact solutions,and four kinds of error results,i.e.,the maximum absolute norm error,L2 norm error,mean root square(MRS)error and average error.The specific features of the software include:1.Software integration is based on the independent difference algorithm.The source codes and theirs'access to the system of each difference algorithm are independent,so that users can locate the algorithms and use them conveniently.This software has good expansibility and compatibility without changing the original software framework.2.Software is based on the cloud platform,which is different from the traditional mathematical calculation softwares required to install stand-alone version or C/S architecture in the local area.It adopts J2EE technology and current mainstream B/S architecture.Only users do a simple configuration on the client,can achieve all operations by sending a request to the server through the browser.3.Software integrates the cloud platform,J2EE framework,MySQL numerical computation language(such as Fortran),mathematical formula,syntax,semantic analysis and other products and technologies.This software contains various technologies and makes them unified for the whole computing platform.4.The system allows the client to call the middleware on the cloud platform through the HTTP request.It can call the popular Matlab and Scilab by the middleware,so as to implement interface with the software Matlab or Scilab.5.The system provides an interface to output a variety of calculation results,which can be read directly or indirectly by users for drawing figures.In addition,the system can draw the graphics directly by using its own drawing plug-in.6.This software combines HTML5,MathJax and LaTex technologies.It is a highlight of the system to implement the layout with rendering and beautiful display of mathematical equations and mathematical formulas on WEB pages.This software achieves numerical solutions of three kinds of PDEs with various finite difference schemes.It is convenient,direct and efficient.With more and more new difference schemes integration,this software will be developed to be a set of open platform for numerical solutions of PDEs.It would be shared and used by all researchers concerning numerical solutions of PDEs in the world.