一维变系数对流扩散方程第三边值问题的紧有限体积方法∗

摘要

Convection diffusion equations have wide applications in engineering computa-tions. In this paper, we study the high accuracy finite volume method for the one-dimensional convection diffusion equation with variable coefficients and third boundary conditions. The integral form of conservation law is derived by integrating the equation over control volumes. Then, the compact finite volume scheme is obtained by discretizing the integral form based on Taylor formula and quadratic Hermite interpolation. The matrix of the deduced linear algebraic system is tridiagonal, which can thus be solved by the Thomas method. Moreover, we analyze the truncation errors and prove that the given scheme is convergent with fourth order accuracy with respect to some standard discrete norms by using the energy method. At last, we provide a numerical example, which demonstrates the correctness and effectiveness of the proposed scheme. It is consistent with the theoretical analysis.%对流扩散方程在工程计算中具有广泛应用。本文研究一维变系数对流扩散方程第三边值问题的高精度有限体积方法。通过在控制体积上积分导出了方程的积分守恒形式，然后对积分守恒形式利用泰勒公式和二次埃尔米特插值进行离散得到了紧有限体积格式。该格式导出的线性代数方程组具有三对角性质，因此可使用追赶法求解。进而，通过分析截断误差，采用能量方法证明了格式按照几种标准的离散范数四阶收敛。最后，数值算例验证了格式的正确性和有效性，这与理论分析结果是一致的。