Bicompact Schemes for the Multidimensional Convection–Diffusion Equation

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.