Abstract

Bicompact schemes are generalized for the first time to the linear multidimensional convection–diffusion equation. Schemes are constructed using the method of lines, the finite-volume method, and bi- and tricubic Hermite interpolation of the sought function in a cell. Time stepping is based on diagonally implicit Runge–Kutta methods. The proposed bicompact schemes are unconditionally stable, conservative, and fourth-order accurate in space for sufficiently smooth solutions. The constructed schemes are implemented by applying an efficient iterative method based on approximate factorization of their multidimensional equations. Every iteration of the method is reduced to a set of independent one-dimensional scalar two-point Gaussian eliminations. Several stationary and nonstationary exact solutions are used to demonstrate the high-order convergence of the developed schemes and the fast convergence of their iterative implementation. Advantages of bicompact schemes as compared with Galerkin-type finite-element schemes are discussed.

中文翻译：

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多维对流-扩散方程的双紧方案

摘要

Bicompact 方案首次推广到线性多维对流扩散方程。使用线法、有限体积法以及单元中所求函数的双和三次 Hermite 插值来构建方案。时间步进基于对角隐式 Runge-Kutta 方法。所提出的双紧方案在空间上是无条件稳定、保守和四阶精确的，以获得足够平滑的解决方案。构建的方案是通过应用基于多维方程近似分解的有效迭代方法来实现的。该方法的每次迭代都被简化为一组独立的一维标量两点高斯消元。几个平稳的和非平稳的精确解被用来证明所开发方案的高阶收敛性及其迭代实现的快速收敛性。讨论了双紧凑方案与 Galerkin 型有限元方案相比的优点。