Информационная система "Конференции"

Тезисы докладов

Численное решение дифференциальных и интегральных уравнений

Hyperbolic systems of equations are basic for the description of wide range of problems in physics and mechanics. Because of nonlinearity, analytical solutions for these equations can not be found except for a small number of special cases. Therefore finite difference algorithms were developed for solution of these systems. Explicit algorithms are easy to implement, however stability constraints make them very expensive in terms of computational cost. The implementation of implicit schemes by direct methods requires inversion of matrices of high dimensions that is also expensive. In recent decades more economical approximate factorization or splitting methods were developed and used extensively. Approximate factorization allows to reduce a problem to a set of simpler problems that can be solved using cheap algorithms such as scalar sweeps. Unfortunately approximate factorization leads to schemes that have lower order of approximation compared to unfactorized schemes and as a result lower precision of calculation or slower convergence for relaxation problems. Exact factorization schemes are not affected by these problems. They are as precise as initial unfactorized schemes and can be implemented economically using scalar sweeps, like approximate factorization schemes. However finding such schemes presents a problem of it's own. Direct undetermined coefficients approach leads to algebraic equations of high order, that not always can be solved. Because of this difficulties it was not known whether precise factorization schemes exist for gas dynamics equations in conservative variables or variables density, pressure, velocity. We propose a new method that allows to find all the exact factorization schemes for any system of one-dimensional hyperbolic equations and determine whether there are ones that can be implemented using scalar sweeps among them. Proposed method was applied to gas dynamics equations in several variables. Exact factorization schemes that can be implemented using scalar sweeps were found for variables density, momentum, pressure, variables density, momentum, momentum flux and variables density, momentum, sonic speed. It was proven that no exact factorization schemes that can be implemented exist for conservative variables, variables density, pressure, velocity and variables density, pressure, sonic speed.

Примечание. Тезисы докладов публикуются в авторской редакции

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