高维流体动力学中若干数学模型的小波数值方法研究

In this paper, using the special space structure of reproducing kernel space, we.establish multi-resolution analysis of differential operator spline and construct the desirable basis of the space with reproducing kernel scale function. At the same time, the decomposition and reconstruction formulae of the functions in the space are established. Combining reproducing.kernel scale function with finite difference method, we give the numerical method that can solve Euler equations. If we adopt the reproducing kernel scale function that has higher regularity, then.the numerical method will has higher precision in the process of solving equations. Numerical tests show that this method has high resolution in shockwave problem and its stability is good enough that can eliminate the phenomena of non-physics oscillation. The computation can be simplified because the epresenting coefficient matrix of the.differential operator under the wavelet basis is quasi-diagonal. Making use of this feature, we can give the wavelet explicit discrete scheme of heat-conduction equation and can make the scheme be suitable for the problem with singular solution by adjusting the magnitude of scale due to the local.properties of wavelet. Error estimate formula and numerical test show that this scheme has higher precision. Furthermore, we give the wavelet interpolation method to solve multidimensional incompressible time-dependent N-S equations of viscous fluid field. We also give the domain.decomposition adaptive algorithm with wavelet method to trace shockwave, which uses stable coupling scheme to process interior boundary and ensures coupling stability of the whole scheme. The pesedospectral-multiwavelet-Galerkin method of the advection-diffusion equations is presented. Numerical test show that this method can process the general boundary conditions and combine the finite difference method with wavelet analysis in the numerical simulation of fluid equations, which can trace the development of the solution better and bring the vantage in the computation and locally oddness problem.

对高维流体动力学中若干数学模型，构造与其相容的高维小波及多分辩分析，寻找处理奇性和线化非线性的途径，使给出的小波数值求解方法能高分辩地捕捉激波处理奇性，达到稳定性好、精度高、计算量小、内存少、适于微机运算的特点，并对其做数值分析系统研究，给出具有较高概括性和应用价值的成果。这是创新工作，其优势是传统数值方法无法比拟的。