【摘要】 本文研究具有Robin边界条件的一维椭圆和抛物型微分方程在非均匀网格上的高阶紧有限体积法.通过对方程等价积分形式的离散化,得到了显式格式和隐式格式.对于基于节点值的显式格式,其对应的线性代数方程组可通过Thomas算法求解;对于同时包含节点值及其导数的隐式格式,采用预测-校正方法实现,其中在校正阶段引入了恢复节点导数值的隐式公式.以两点边值问题为例,利用能量方法证明了显式与隐式格式在离散范数下均具有四阶收敛精度.两个数值算例验证了格式的正确性与有效性,同时表明了采用非均匀网格的必要性.更多还原

【Abstract】 This paper studies high order compact finite volume methods on non-uniform meshes for one-dimensional elliptic and parabolic differential equations with the Robin boundary conditions. An explicit scheme and an implicit scheme are obtained by discretizing the equivalent integral form of the equation. For the explicit scheme with nodal values, the algebraic system can be solved by the Thomas method. For the implicit scheme with both nodal values and their derivatives, the system can be implemented by a prediction-correction procedure, where in the correction stage, an implicit formula for recovering the nodal derivatives is introduced. Taking two point boundary value problem as an example, we prove that both the explicit and implicit schemes are convergent with fourth order accuracy with respect to some standard discrete norms using the energy method. Two numerical examples demonstrate the correctness and effectiveness of the schemes, as well as the indispensability of using non-uniform meshes.更多还原