Optimal monotonization of a high-order accurate bicompact scheme for the nonstationary multidimensional transport equation

Two variants of parallel implementation of high-order accurate bicompact schemes for multi-dimensional inhomogeneous transport equation

“…Without loss of generality, the basic points of the method used to construct multidimensional bicompact schemes [6,7] can be described as applied to the threedimensional nonstationary transport equation 3 3…”

“…Since only a finite difference of the primitive was involved in the scheme in [4], starting from [5], the bicompact scheme was written in a computationally more convenient form: the finite differences of the primitive were replaced by integral averages of the sought function over grid cells. Later, bicompact schemes were constructed for the two-dimensional (2D) and three-dimensional (3D) nonstationary inhomogeneous linear transport equations [6,7] and for systems of nonstationary multidimensional quasilinear hyperbolic equations [8]. Note that, in bicompact schemes [8,9] for quasilinear hyperbolic equations, an increase in the order of accuracy is ensured not with the help of auxiliary integral averages of the sought function over grid cells, but rather with the help of auxiliary values of the sought function at half-integer spatial grid nodes.…”

“…To find these quantities, additional difference equations are derived that approximate consequences of the basic differential equations. The equations of semidiscrete bicompact schemes are usually integrated with respect to time using A-and L-stable diagonally implicit Runge-Kutta (DIRK) methods [4][5][6][7][8]. In the case of smooth solutions, the important advantages of bicompact schemes include the preservation of their order of accuracy on an arbitrary nonuniform spatial grid and the low cost of the marching method used to solve the equations of the scheme in each spatial variable.…”

“…In the case of smooth solutions, the important advantages of bicompact schemes include the preservation of their order of accuracy on an arbitrary nonuniform spatial grid and the low cost of the marching method used to solve the equations of the scheme in each spatial variable. In [6,8] high-order accurate bicompact schemes were used to construct hybrid schemes for computing discontinuous solutions of hyperbolic equations.…”

“…For efficient computations of the nonstationary multidimensional inhomogeneous transport equation, a parallel space marching algorithm for computing difference equations of high-order accurate bicompact schemes was proposed in [6]. In this paper, we consider another parallel algorithm for solving the difference equations of high-order accurate bicompact schemes.…”