1维非定常对流扩散方程的有理型高阶紧致差分格式

A Rational High-Order Compact Difference Scheme for the 1D Unsteady Convection-Diffusion Equation

针对1维非定常对流扩散方程,首先建立了1种2层有理型高阶紧致差分格式,其局部截断误差为O(h4+τ2).然后采用von Neumann分析方法证明了该格式是无条件稳定的.由于在每个时间层上只涉及到3个网格点,因此可直接采用追赶法求解此差分方程.最后通过3个数值算例验证了方法的精确性和可靠性.数值结果表明:所述格式不仅能够适用于非定常对流扩散问题,而且能够较好地求解非定常纯对流问题或纯扩散问题,并且其计算效果均优于Crank-Nicolson(C-N)格式和指数型高阶紧致(EHOC)差分格式.

A two-level rational high-order compact difference scheme for solving the 1D unsteady convection-diffu-sion equation is proposed. The local truncation error of the scheme is O（h4 ＋ τ2 ）. It is proved that is unconditionally stable by von Neumann analysis method. Because only three points are used at each time level,this difference scheme can be solved by the method of forward elimination and backward substitution. Finally,numerical experi-ments for three test examples are carried out to demonstrate the accuracy and the effectiveness of the present meth-od. It is found that the present method is not only easy to be implemented to solve the 1D unsteady convection- dif-fusion problems,but also can be used to solve the unsteady pure convection problems or the pure diffusion prob-lems. And the computed results are better than Crank-Nicolson（C-N）scheme and the exponential high-order com-pact（EHOC）difference scheme.