Combined Method For A Class Of Patitioned Large Stiff Nonautonomous Systems

Stiff differencial equations(SDE) always have it's applications in high-tech scopes.like aviation, thermic nuclear reaction, automatic control, electronic network & chemical dynamics etc., which have great importance to do with national defence and modernization construction.So, it goes without saying to be very important.Been discreted by space variable, partial differencial equation(PDE) with initial value or border value would be converted to big scale ordenary differencial equations(ODE),and it is an important headspring that causes stiff problem.For linear stiff problem ,the main theory have A-stability(Dahlquist,1963),A(α)-stability(Widlund,19 67) and stiff stability (Gear,1969).There were many demestic scholar searched linear theory,including Jiaoxun Kuang, Guoliang Chen.Jiaxiang Xiang,Shoufo Li,Chenming Huang,Qingyang Li,Jinggao Fei.To research for non-linear stiff problem,Butcher & Burrage established B-stable and algebra stable theory for Runge-Kutta method from 1975-1979(see[17][19][21~ 23] etc.).Frank, Schneid & Ueberhuber established B-convergence theory for Runge-Kutta method from 1981-1985(see[14][15]etc). At the same time, there are many demestic scholar dedicate to B therom(see[8~10]etc.).But the research about convergence of nonlinear stiff problem didn't reach ideal aim that can be analysed to ration convergence. In recent years, people pay attention to extensive ordinary multivalue methods more and more. Mix method is one of popular method in recent years. And to enhance calculation speed of stiff system ,many scholar use parallel methed(see[2][9][20]etc). Merson, Butcher, Hairer & Wanner etc. invented root tree theory(see[4] [5] [6]etc). Just from root tree theory , mix method and parallel method,this paper gives combined RK-Rosenbrock method, which can parallel realize a certain stiff non-autonomy system numerical solution. This paper is the development of RK-Rosenbrock method of professor Chen Lirong. There are some details about combined RK-Rosenbrock method of stiff autonomy large system in bibliography [1][2][3].The first part of this paper is going to introduce root tree theory of forerunners; The second part is going to give combined RK-Rosenbrock method, and discuss it's rank condition; The third part of this paper will check A-stable conditions of the stiff part in equation system; The fourth part will give convergence theorm and stabilization theorm;The fifth part will give one numerical example , calculated result and the result analsis; My appendix will be filled with exact program , that is my main result which put theory into practice.