With the solution reconstructed in this way, the numerical errors accumulate in the fashion that the local truncation errors in two successive time steps cancel each other, and this leads to the second-order accuracy of the scheme. In doing so, the smooth assumption on the solution is not necessary. The scheme then uses them to reconstruct the solution in the cell by enforcing the algebraic relations among them, with certain TVD limiting to maintain the stability. Different from the above approach, our scheme numerically computes more physical quantities, which are algebraically related with one another in each cell. Different limiting technologies, such as TVD, ENO, and WENO, are then used to eliminate numerical oscillations caused by the presence of discontinuities. The schemes so developed are no more local as the original Godunov scheme. Traditional ways to extend it to high-order schemes are to use high-order interpolations in the solution reconstruction in each cell, assuming that the solution is smooth. The original Godunov scheme is first-order accurate. The significance of the entropy scheme is in methodology. The entropy scheme was extended to the Euler system in. generalize the entropy-TVD scheme for the one-dimensional shallow water equations in. Furthermore, the scheme suits for long-time numerical computing. The scheme is second-order, but the numerical results showed that the scheme has a third-order convergence rate away from extrema. In, Cui and Mao extended the entropy scheme to the KdV equation. In, Chen and Mao extended the entropy scheme to the nonlinear scalar conservation laws and presented the entropy-TVD scheme.
In essence, entropy-ultra-bee scheme is a combination of the entropy scheme and the ultra-bee scheme which can obtain good resolution in smooth regions and sharpen the discontinuity. In order to eliminate the spurious oscillations, an entropy-ultra-bee scheme was presented by Li and Mao for computing the linear advection equation. However, when computing discontinuous solutions, spurious oscillations occurred in the vicinity of the discontinuities. Yanfen and De-Kang investigated the truncation error for the entropy scheme and showed the entropy scheme has superconvergent property in. The numerical tests showed that it can achieve very good accuracy and is suitable for long-time computation of smooth solutions. In, Li has developed the entropy scheme which contains numerical solution and numerical entropy to compute the linear advection equation. Many researchers have developed numerical methods for the convection-diffusion equation and have obtained some superconvergence results. In this paper, we consider the convection-diffusion equation: where. Numerical tests show that the error achieves about second-order accuracy, but the error reaches about forth-order accuracy. The operator splitting method is used to solve the convection-diffusion equation that is divided into conservation and diffusion parts, in which the first-order accurate entropy scheme is applied to solve the conservation part and the second accurate central difference scheme is applied to solve the diffusion part. In this paper, we extend the entropy scheme for hyperbolic conservation laws to one-dimensional convection-diffusion equation.