Abstract
This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving one-dimensional advection-diffusion equation (ADE). Variational formulation of this type equation and corresponding numerical fluxes are devised by utilizing the advantages of both the Legendre wavelet bases and discontinuous Galerkin (DG) method. The distinctive features of the proposed method are its simple applicability for a variety of boundary conditions and able to effectively approximate the solution of PDEs with less storage space and execution. The results of a numerical experiment are provided to verify the efficiency of the designed new technique.
Highlights
The advection-diffusion equation arises in many important applications, such as fluid dynamics, heat transfer and mass transfer etc. [1]-[9]
The above nice properties demonstrate that the Legendre wavelet bases can be very efficiently applied to the numerical solution of Equation (1)
The transition matrices have large dimensions 2n p × 2n p, they have block diagonal structures. We can adopt another calculation technique of the differential operator, i.e., computing on each element Inl other than whole interval [0,1]. This approach consists of the discontinuous Galerkin (DG) method for solving the partial differential equations (PDEs) and avoids the discontinuity of the Legendre wavelet bases at interfaces
Summary
The advection-diffusion equation arises in many important applications, such as fluid dynamics, heat transfer and mass transfer etc. [1]-[9]. The reason for such fast development of the WG approach may be the fact that many nonlinear PDEs have solutions containing local phenomena and interactions among several scales, which can be well-represented in wavelet bases owing to their nice properties, such as compact support and vanishing moment [15]-[22] Their main limitations are the difficulties to adapt them to non-periodic geometries and to append specific boundary conditions. The DLWG approach utilizes the discontinuous feature at nodes of the Legendre wavelet bases combined with discontinuous finite elements to discretize the space variable and the spacial derivatives to produce a system of first-order ODEs in time for Equation (1.1) We solve this system by using the TVD Runge-Kutta method [11], and obtain good numerical results, illustrating that this scheme is very simple and computationally efficient. Conclusions of the proposed method and some suggestions for future research are given at the end of Section 7
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