The Explicit/Implicit Euler Scheme For The Conduction Convection Equations

The fluid flow is governed by the nonlinear equations, so it is very difficulty to study the behavior of fluid flow. If heat transfer associates with fluid flow, it would be governed by the fluid thermal dynamical system, which is gotten by the Navier-Stokes equations coupled with the energy equation through the Boussinesq approximation. In the process of numerical calculation, there are all the difficulties like calculating the Navier-Stokes equations. In fact, because of the viscosity of the fluid it will inevitably produce heat in the process of fluid flow. Therefore, there are interaction and transformation of the speed, pressure, and temperature in this process. Hence, the study of the fluid thermal dynamical system has important theoretical and practical signification, which can help us better understand the universal law of fluid flow.In this dissertation, we study the explicit/implicit Euler scheme for the conduction convection equations, the main research contents are as follows:Firstly, we set up the space-discrete numerical scheme of the unsteady heat conduction convection equations based on variation principle, mixed finite element approximation theory. The stability and the corresponding convergence are provided, and the corresponding optimal error estimates for velocity and temperature in2 L norm are established. Theoretical analysis shows that the algorithm is stable and has a good numerical accuracy. Finally, some numerical experiments are provided to verify the correctness of the established theoretical results.Secondly, due to the discrete numerical scheme is a nonlinear system, It is very difficult to find the numerical solution of such problem. In order to simplify the computation, we adopt the implicit scheme to treat the linear term and explicit scheme to deal with the nonlinear term. Namely, we establish the explicit/implicit Euler scheme for the conduction convection equations, in this way, we not only reduce the amount of calculation, but also keep the good accuracy. As a consequence, the nonlinear numerical scheme is changed into a linear problem, the conduction convection problem is solved effectively.Finally, We present the stability of the explicit/implicit Euler scheme and develop the optimal error estimates for velocity and temperature in different norms. Compared with the classical literature, some numerical tests are given to comfirm the superiority of the our new algorithm and the correctness of the theoretical resultss.