两点周期边值问题的紧有限体积方法

Compact finite volume scheme for two point periodic boundary value problem

针对两点周期边值问题提出了一种紧有限体积格式,该格式形成的线性代数方程组具有周期三对角性质,通过变换,将其变为2个三对角方程组,使用追赶法求解,提高了计算效率.利用能量方法证明了格式按照H1半范数和L2范数具有四阶收敛精度,并给出了单元中点值和一阶导数值的高精度后处理计算公式,得到其具有四阶精度.数值算例验证了理论分析的正确性和格式的有效性.

A compact finite volume scheme is presented for two point periodic boundary value problem. The linear algebraic system derived by this scheme has periodic tridiagonal property. By eonsructing a transformation, two linear algebraic systems which have tridiagonal property are obtained and can be solved by Thomas method. It is proved that the given scheme is con- vergent with fourth order accuracy with respect to discrete H1 semi-norm and L2 norm by energy method. Furthermore, the post-processing formulae for the numerical value and derivative at the midpoint of every element are obtained, which have fourth order accuracy. Numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme and its extrapolation.