Improvement Of Nine-point Difference Scheme For Two-dimension Three-temperature Heat Conduction Equations

Inertial confinement fusion is an important way for achieving thermonuclear fusion. The research in this field is of great importance since it will provide clean energy for the development of national economy. Numerical simulation is a very important aspect for this research, in which the main task is to solve the system of radiation hydrodynamic equations, especially, we usually do this task using Lagrangian coordinates, and in this case the numerical treatment of three-temperature energy equations is a main difficulty. In order to solve two-dimension three-temperature energy equations over an irregular quadrangle grid, a lot of researches have been done in literature, and several different approaches have been proposed, such as five-point difference scheme, nine-point difference scheme, flux iteration scheme, finite element scheme, etc. Up to now, the nine-point difference scheme is in general recognized as the better one and is used more generally.The main goal of this paper is to make a thorough research for the nine-point difference scheme, make clear its advantages and disadvantages, and improve it in view to its main disadvantages to raise the computational accuracy, speed and efficiency. The main results obtained in this paper are as follows:(1) It is discovered that the nine-point difference scheme is not consistent over any irregular quadrangle grid except that the grid is consists of some congruent parallelograms. In fact, we have done theoretical analysis and a series of numerical experiments, which show that for the nine-point scheme applied to a system of two-dimension three-temperature equations over an irregular quadrangle grid, both the local accuracy and global accuracy are very disappointing. This means that the nine-point scheme does not apply to irregular quadrangle grids. Moreover, we also find that the nine-point scheme does not apply to heat conduction equations with diffusion coefficient drastically varying. In fact, the the nine-point scheme does not work well, even does not work completely, when it is applied to any given heat conduction equation of the above type. It's a surprising that such a well-known difference scheme actually has the above drawbacks which have never been pointed out in any publication.(2) An improving program is brought forward to overcome the aforementioned drawbacks of the nine-point scheme, and an improved nine-point scheme is constructed. Theoretical analysis and numerical experiments show that the improved nine-point scheme not only has completely overcome the second drawback mentioned above that the original nine-point scheme can not apply to the parabolic problem with diffusion coefficient drastically varying, but also has partly improved the applicability of the scheme when performed over irregular grids.(3) We bring BDF method of order two forward as time discrete method for the two-dimension three-temperature heat conduction equations. Theoretical analysis and numer-ical experiments show that BDF method of order two is indeed superior to the commonly used backward Euler method and Crank-Nicolson scheme.(4) To raise the computational accuracy and efficiency, we have designed a self-adaptive space grid that can guarantee mass conservation and energy conservation, and have also developed the techniques for the time stepsize automatically varying. Based on these new techniques, the software with automatically variable irregular space grid and automatically variable time stepsize has been developed, which greatly enhances the computational efficiency for the numerical solution of two-dimension three-temperature equations.