Институт вычислительной математики
и математической геофизики

Abstracts

Numerical solution of differential and integral equations

New splitting methods (decomposition, fractional steps, and alternating directions) are proposed for partial differential evolutionary equatons. The development of these methods is based on diagonal implicit methods. For these methods to have the highest attainable order of accuracy, their coefficients must satisfy both previously known and certain additional conditions. The coefficients found by using a certain technique that optimally takes into account stability and accuracy requirements.

Assume that, in evolutionary equation $u_t=Au+f(t)$ the differential operator $A$ is the sum of operators $A_{i},quad i=1,dots,p$. Consider also a different decomposition $A=sum_{i=1}^s A_{i}^o$ $(sge p).$ For example, if $A=A_1+A_2$, and $s=4$ then $A_1^o=A_3^o=A_1/2$ and $A_2^o=A_4^o=A_2$. We suggest a new $s$-stage implicit diagonally Runge-Kutta method

$$ u_{n+1}=u_{n}+ au sum_{i=1}^s b_{i} left(Ay_i+f_i ight), $$

$$ y_{i}=u_{n}+ au sum_{j=1}^{i-1} a_{ij}left(Ay_j+f_j ight)+ au a_{ii} left(A_i^o y_i+g_i+f_i ight), $$

where

$$ f_i=fleft(t_n+c_i au ight),quad g_1=sum_{j=2}^s A_j^o u,quad g_i=sum_{j=1,j e i}^s A_j^o y_j(i>1). $$

It is possible to find coefficients of methods for $p=2, s=2,3,4$, $p=3, s=2,3$. The test computations provide better results as compared to the relaxation and predictor-correct methods. The relaxation and predictor-correct methods are advantageous in that their implementation requires fewer arithmetic operations; however, they provide good resultsonly when the time step is small.

References

1.Shirobokov, N.V., Diagonal Implicit Runge--Kutta Scemes. Zh. Vychisl. Mat. Mat. Fiz., 2002, vol. 42, no. 7, pp. 1013--1018.

2.Shirobokov, N.V., Evolutionary Equations Spliting on the Basis Implicit Methods. Zh. Vychisl. Mat. Mat. Fiz., 2003, vol. 43, no. 9, pp. 1416--1423.

Note. Abstracts are published in author's edition

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