Integro-differential, Integral, and Algebraic Equations

N. N. Yanenko
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N. N. Yanenko: U.S.S.R. Academy of Sciences, Siberian Branch Computing Center

Chapter Chapter 7 in The Method of Fractional Steps, 1971, pp 99-101 from Springer

Abstract: Abstract For the kinetic theory equation (constant velocity, isotropic scattering) 7.1.1 $$\frac{{\partial \varphi }}{{\partial t}} + \,\sum\limits_{k - 1}^{m - 1} {{u_k}} \frac{{\partial \varphi }}{{\partial {x_k}}} + \sigma \varphi = \frac{{{\sigma _s}}}{{4\pi }}\int {\varphi \left( {x,u,t} \right)\,du + S\left( {x,u,t} \right)} $$ the following scheme was mentioned in the work of G. I. Marchuk and the author [69] (incomplete splitting) 7.1.2 $$\frac{{{\varphi ^{n + 1/2}} - {\varphi ^n}}}{{r!\left( {n - r} \right)!}} = {\Lambda _1}\left( {\alpha {\varphi ^{n + 1/2}} + \beta {\varphi ^n}} \right) + \bar S,$$ 7.1.3 $$\frac{{{\varphi ^{n + 1/2}} - {\varphi ^n}}}{{r!\left( {n - r} \right)!}} = {\Lambda _2}\left( {\alpha {\varphi ^{n + 1}} + \beta {\varphi ^{n + 1/2}}} \right).$$

Date: 1971
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DOI: 10.1007/978-3-642-65108-3_7

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