Wavelet Methods For The Numerical Solution Of Hyperbolic Equations

The rapid development of science and technology brings increasing interest in numerical computation.As the magnitude of problems studied in fluid dynamics becomes larger,there is an urgent requirement in practice that computational mathematics should provide accurate,efficient and easy-use numerical tools to simulate and analyse the flow-fields.It is well known that the exact solution to such equations may develop singularities in fairly localized regions,such as shocks and bow waves, which brings essential difficulties to classical mathematics.Since solution changes acutely near singularities,it is not practical to use regular meshes and linear PDE solver. Recently,wavelet,as a powerful adaptive tool,attracts much attention of researchers in computational fluid dynamics.In addition to applying widely to engineering areas such as signal processing,it becomes gradually an alternative numerical method for the solution of partial differential equations.With a view to discontinuities simulation,this paper considers wavelet methods for hyperbolic equations,especially hyperbolic conservation laws.Its objectives are twofold.The first is to discuss how wavelet can be used to simulate discontinuities,the second is to discuss how wavelet analysis can be introduced to improve computational efficiency of traditional methods.Our main work includes:1.By using Daubechies scaling functions as weight and test functions of Galerkin method,we construct and analyze two classes of wavelet Galerkin schemes for Hamilton-Jacobi equations.Due to the localization of wavelet in time and frequency,the new schemes can restrain numerical oscillations effiently and are suitable for problems with singularities.Its feasibility is proved by numerical results.2.Based on the equivalent relation between Hamilton-Jacobi equations and hyperbolic conservation laws for the one-dimensional case-the viscosity solutions are equivalent to the entropy solutions,we adopt firstly the integral transformation so that hyperbolic conservation laws are transformed into their counterparts and then use wavelet Galerkin methods that designed for Hamilton-Jacobi equations to solve them. Since solutions of the underlying Hamilton-Jacobi equations are of Lipschitz continuity, this approach avoids applying wavelet directly to approximate strong discontinuities and the well-known Gibbs phenomena are eliminated.Numerical results show that these new methods are effective. 3.A class of adaptive multiresolution schemes are constructed for the systems of hyperbolic conservation laws.The rough idea is to use a hierarchy of fixed nested grids at different resolution,which offers the possibility of locally selecting an appropriate level of discretization.By choosing suitable threshold,the computational grids will concentrate automatically in regions where strong gradients of numerical solutions are observed.Then,the differential equations are solved by high-resolution finite volume schemes(for example,TVD and ENO,etc.) on these adaptive grids.Since finite volume method is inherently conservative,the total mass conservation can obtained by introducing a local flux correction at grid interface.Therefore,the numerical solutions can converge to the weak solution of the underlying hyperbolic system.On the other hand,since the wavelet coefficients can reflect local smoothness of the numerical solution,the grids will adapt dynamically and total sparsity can be controled by magnitude of the filtering threshold.Numerical experiments indicate that,despite some loss of accuracy,our approach is excellent at improve efficiency and can be viewed as a good substitution of the moving mesh method.4.Considering the fact that many existing wavelet collocation methods are using explicit multistage temporal discretization,we discuss semi-implicit wavelet collocation method for convection diffusion equations.To preserves the high-accuracy of wavelet approximation and high-resolution to singularities,spatial derivatives are discretized by the autocorrelation functions of Daubechies wavelet;while the Crank-Nicolson method is adopted to deal with the temporal discretization.Among all two-point difference schemes,Crank-Nicolson scheme is the one which has the highest order.Furthermore,it bears many advantageous properties,such as simple and efficient.Due to the localization of wavelet functions,the algebraic system arising from the collocation scheme is band and sparse matrix.Finally we use Von Neumann method to analyse and prove that the scheme is absolutely stable for the autocorrelation functions of Daubechies wavelet that used in this paper.