Of And Application Of Multiscale Computational Methods

Many problems in science and engineering take on multiscale characteristic. For these problems, traditional numerical methods are hard to solve them for huge calculation cost. Therefore, multiscale computational methods are used to solve multiscale problems, because these methods can not only save calculation cost but also keep calculation precision. At the present time, some classical multiscale computational methods have appeared, such as multi-grid method, homogenization, waved-based numerical homogenization, multiscale finite element method and the heterogeneous multiscale method etc. These methods are successfully applied in science and engineering. In this paper, homogenization method, the heterogeneous multiscale method and wave-based numerical homogenization are researched. The main results are as follows:Firstly, based on the framework of the heterogeneous multiscale method, the finite volume heterogeneous multiscale method is proposed for multiscale hyperbolic equations. The numerical method relies on two different schemes of original equation, and gives numerical results at a low cost by coupling microscale scheme to macroscale scheme. Numerical results show that the calculation cost of the proposed method is less than that of the finite volume method.Secondly, wave-based numerical homogenization is applied to solve parabolic equations with highly oscillating coefficients. Elliptic equations have been solved successfully by wavelet-based numerical homogenization, but it has not been applied to solve parabolic equations. In this paper, wave-based numerical homogenization is applied to solve parabolic equations with highly oscillating coefficients. Numerical results show that this method is better than the finite volume method, not only in calculation cost but also in accuracy.Thirdly, wave-based numerical homogenization is improved for flow problems in heterogeneous porous media. Algorithmic precision is improved by error correction. Numerical results show that improved wavelet-based method is better than the traditional finite volume method, not only in computational cost but also in accuracy. Compared with wavelet-based homogenization method, calculation accuracy of the improved method is improved.Finally, wave-based numerical homogenization is applied to simulate ground-water flow problem. Groundwater system in nature is almost heterogeneous, and themethod is applied to solve unsteady flow problems in heterogeneous porous media with three kinds of parameters which are continuous, abrupt and locally oscillating. The computational results show that wavelet-based method is better than the traditional finite difference method not only in cost but also in accuracy.