The Multi-domain Chebyshev-Legendre Spectral Methods For The Two-dimensional Vorticity Equations

Spectral method is an important method for solving partial diferential equations.Thehigh order of convergence rate for problems with smooth solutions is its biggest advan-tage.But the spectral method has difculty in solving equations with complex area.Whilethe multi-domain spectral methods maybe solve this problem efciently. Especially,by themulti-domain spectral methods,a complex problem can be partitioned into several smallproblems. Generally,the multi-domain method is an efcient technique to improve the op-erating rate because we can make parallel computation. The non-conformance and sin-gularity of the Chebyshev weight function maybe result in the Chebyshev approximationscheme is not stable.For the multi-domain spectral methods,the singular weight function onthe boundary will make this problem more serious.Although the Legendre spectral methodhas not weight function,but it is lack of efective fast transfer method. The Legendre col-location points and weight are implicit type.The Chebyshev-Legendre spectral method notonly avoid these shortcomings but also make good use of their advantages such as lesscomputations,simple scheme, its easier to be generalized to multi-domain approaches.In this thesis, we study the multi-domain Chebyshev-Legendre spectral methods forthe two-dimensional vorticity equations.Firstly,we introduce some properties which are essential in numerical analysis ofChebyshev interpolation operator in two-dimensions with multi-domain in general L2-norm.Secondly,we apply a multi-domain Legendre Galerkin Chebyshev collocationmethod(MLGCC) to the two-dimensional vorticity equations.The multi-domain spectralmethods are”splicing methods”. MLGCC means the LGCC methods are applied on allthe subintervals. The LGCC scheme is basically formulated in the Legendre Galerkin formbut with the nonlinear term being treated with the Chebyshev collocation method. TheChebyshev-Gauss points are adopted. And it needs less computing time with the help of theChebyshev-Legendre transform. We introduce appropriate base functions so that the matrixof the equations are sparse and describe the parallel algorithm of the MLGCC method.At last,we introduce some approximation properties of Chebyshev interpolation op-erator. The stability and the optimal rate of convergence of the method are proved.Theerror estimation is obtained in L2-norm.Numerical results are given for both single domainand multi-domain methods to make a comparison. The validity of the domain decompo-sition Chybeshev-Legendre spectral method and the correctness of the theoretical analysisare observed.