Finite Volume Weno Schemes For Parabolic Equations And A Class Of Nonlinear Evolution Equations

Many the problems of modern science and engineering can be described by partial differential equations. Since quite a lot of these problems are time-dependent, most of them are evolution equations and especially nonlinear evolution equations Due to the complexity of the real problems, it is very hard to obtain the exact solutions or its exact solution is too complex to have no use. As a result, people have to look for numerical methods for these equations. The finite volume weighted essentially non-oscillatory(WENO) schemes are the useful and high-degree method [5].The finite volume WENO schemes are new and high-degree computing methods for partial differential equations. At present, there are many research to design the high-degree and high-resolution of conservation laws equations and Hamilton-Jacobi equations by those schemes [2-11], while there are less research for parabolic equations and some nonlinear evolution equations, so this paper will use the finite volume WENO schemes for parabolic equations and a class of nonlinear evolution equations to study the numerical solution. Our main contributions of this paper are as follows:(1) In the second and third chapter, finite volume WENO schemes are applied to study one-dimensional and two-dimensional parabolic equations, and obtain a discretization schemes of parabolic equations. Through the corresponding numerical examples and by absolute error, relative error of analysis to verify the validity of the format. (2) In the fourth chapter, finite volume WENO schemes are applied to study a class of nonlinear evolution equations. We also obtain a discretization scheme of the nonlinear evolution, give several numerical examples, and the corresponding analysis of effectiveness is verified.