High-order Alternating Segment Explicit And Implicit Scheme For Burgers Equation

Burgers equation has some properties of the Navier-Stokes equation, it can be used as a mathematical model of a kind of fluid flow phenomena, we can get academic value of the equation other than the value of the solu-tion for the equation itself, and the parallel numerical solution of Burgers equation is concerned with the development of parallel computing. Even the finite difference method and finite element method for Burgers equation has been developed in recent decades, the construction of numerical method for Burgers equation which is high-order accuracy and suitable for parallel using is still a subject of people's attention.We give a set of alternating group explicit and implicit scheme for Burgers equation. In the process of numerical algorithm, we give the higher order explicit and implicit schemes for Burgers equations. Then we con-struct four forms of higher order non-symmetric format based on the implicit and implicit schemes. When the Burgers equation has periodic boundary conditions, first of all, we construct a class of alternating group four-point difference method using these four high-order alternating segment explicit and implicit non-symmetric form, and we can get the corresponding matrix format. Then on the base of higher order explicit and implicit difference scheme, using the four non-symmetric format, we structure a class of al-ternating segment explicit and implicit method. In the odd-time level, we use the form of "explicit section, implicit section,...,explicit section, im-plicit section" to construct difference schemes. However, in the even-time level, we use the form of "implicit section, explicit section,..., implicit sec-tion, explicit section". Thus, we get the higher-order alternating group explicit and implicit scheme for Burgers equation.At the same time, we get its corresponding matrix.In the discrete process of Burgers equation, we need to change the nonlinearity items into linear items. We use the value of nodes at the N- layer to approximate the value of nonlinearity items.When the Burgers equation has non-periodic boundary conditions, on the basis of alternating segmented explicit and implicit difference scheme for the periodic boundary conditions, we use the same segmentation method. Particularly, on the left and the right boundary point within the Burgers Equation's discrete space, we use non-symmetric explicit and implicit dif-ference schemes. Thereby, we get the matrix of alternating segment explicit and implicit difference scheme, which is different from the periodic boundary conditions. After the given of Burgers equation's AGE four-point difference method and the alternating segment difference method, we analysis the linear stability of the alternating segment explicit and implicit difference scheme, we prove that the alternating segment difference method is linear absolute stability according to Kellogg lemma given in this article.At the end of article, we give a specific numerical example for Burg-ers equation. Then, in the case of non-periodic boundary conditions, we give different methods to get the numerical approximation. From the nu-merical solution of the numerical example, we can see that the alternating segment difference method has the nature of calculate parallel, and it has high convergence rate which is close to fourth order for the space variable.