|
New numerical method of collocation and least-squares (CLS) was developed for solving problems for one-dimensional nonlinear hyperbolic equation of second order. In this method, the solution in each grid cell is searched for as linear combination of polynomial basis functions. In order to find coefficients of solution expansion by the basis, we write down an overdetermined system of collocation equations derived from requirement that hyperbolic equation, matching conditions at intercell boundaries and boundary conditions are to be satisfied at some points. The solution of this system is then found by least-squares method. As compared with finite-difference schemes, boundary conditions of any type are easily realized, approximation order can be relatively easy increased due to choice of basis functions, numerical solution is known at any point of the domain in the method proposed.
By change of variables the second order equation can be reduced to the system of first order equations with respect to derivatives. In case of solving problems where solution derivatives are discontinuous, CLS method for system of equations resolves the behaviour of derivatives at discontinuities somewhat better than in case of finding them by differentiating the solution obtained by CLS method for single second order equation.
Problem about longitudinal oscillations of a rod with different tension and compression coefficients was considered. In this problem, first derivatives with respect to time and spase have discontinuities, and a shock exists in the domain. Solving this problem by CLS method gave more precise results as compared to Godunov's scheme and TVD scheme with van Leer limiter using Lax-Wendroff approximation of flux in regions with smooth solution and Godunov's approximation on discontinuities.
Mail to Webmasterwww@www-sbras.nsc.ru |
|Home Page| |English Part| |
![[SBRAS]](http://www-sbras.nsc.ru/gif/s_sigma.gif) Go to Home |