B. V. Rogov

Monotonic bicompact schemes for linear transport equations

The biocompact difference scheme earlier proposed by these authors for a linear transport equation, which has the fourth-order approximation in spatial coordinate on the two-point stencil and the first-order approximation in time, is monotonic. This implicit scheme is absolutely stable and can be solved by explicit formulas of a running calculation. On the basis of this scheme a monotone non-linear homogeneous difference scheme of high (third for smooth solutions) order accuracy in time is constructed. Calculations of test problems with discontinuous solutions have demonstrated that the proposed scheme has a significant advantage in accuracy over the known nonoscillatory schemes of high-order approximation.

Monotone high-accuracy compact running scheme for quasi-linear hyperbolic equations

The monotone homogeneous bicompact difference scheme earlier proposed by the authors for the linear transport equation is generalized to the case of quasi-linear hyperbolic equations. The generalized scheme is of the fourth order of approximation in the spatial coordinates on a compact stencil and has the first order of approximation in time. The scheme is conservative, absolutely stable, monotone over a wide range of local Courant number values and can be solved by explicit formulas of the running calculation. A quasi-monotone three-stage scheme, which has the third-order approximation in time for smooth solutions, is constructed on the basis of the scheme with a first-order time approximation. Numerical results are presented demonstrating the accuracy of the proposed schemes and their monotonicity in the solution of test problems for the quasi-linear Hopf equation.

On the convergence of compact difference schemes

Difference schemes that are compact in space, i.e., schemes constructed on a two- or three-point stencil in each spatial direction, are more efficient and convenient for boundary condition formulation than other high-order accurate schemes. Originally, these schemes were developed primarily to obtain smooth solutions. In the last two decades, compact schemes have been actively used to compute gas dynamic flows with shock waves. However, when a numerical solution with guaranteed accuracy is desired, the actual properties of difference schemes have to be known in the calculation of solutions with discontinuities. For some widely used compact schemes, this issue has not yet been well studied. The properties of compact schemes constructed by the method of lines are examined in this paper. An initial-boundary value problem for the linear heat equation with discontinuous initial data is used as a test problem. In the method of lines, the spatial derivative in the heat equation is approximated on a two-point stencil according to a fourth-order accurate compact differentiation formula. The resulting evolution system of ordinary differential equations is solved using various implicit one-step two- and three-stage schemes of the second and third order of accuracy. The relation between the properties of the stability function of a scheme and the spatial monotonicity of the numerical solution is analyzed. In computations over long time intervals, the compact schemes are shown to be superior to traditional schemes based on the second-order accurate three-point approximation of the spatial derivative.

Boundary conditions implementation in bicompact schemes for the linear transport equation

Boundary conditions implementation in previously proposed bicompact schemes is studied. These schemes are constructed by the method of lines for a linear transport equation. These schemes are conservative, monotonic, and economical and can be solved by running calculation method. Methods are proposed for the boundary conditions implementation in bicompact schemes that ensure their high accuracy by using A- and L-stable diagonally implicit Runge-Kutta schemes with the third-order approximation for the time integration of the transport equation.

Bicompact Monotonic Schemes for a Multidimensional Linear Transport Equation

Bicompact difference schemes, previously proposed by the authors for linear one-dimensional transport equations are generalized to the multidimensional case by using a coordinate-wise splitting of the multidimensional problem. The scheme stencil for each of the spatial directions is minimal and consists of two points. The schemes are efficient and can be solved by the running calculation method. The proposed difference schemes have the fourth-order approximation in space variables and first- or third-order time approximation for smooth solutions. The schemes for solving multidimensional problems have inherited the monotonicity property of one-dimensional bicompact schemes. Numerical examples are given illustrating the actual accuracy order of bicompact schemes for smooth solutions and the scheme monotonicity for discontinuous solutions.

Minimal dissipation hybrid bicompact schemes for hyperbolic equations

New monotonicity-preserving hybrid schemes are proposed for multidimensional hyperbolic equations. They are convex combinations of high-order accurate central bicompact schemes and upwind schemes of first-order accuracy in time and space. The weighting coefficients in these combinations depend on the local difference between the solutions produced by the high- and low-order accurate schemes at the current space-time point. The bicompact schemes are third-order accurate in time, while having the fourth order of accuracy and the first difference order in space. At every time level, they can be solved by marching in each spatial variable without using spatial splitting. The upwind schemes have minimal dissipation among all monotone schemes constructed on a minimum space-time stencil. The hybrid schemes constructed has been successfully tested as applied to a number of two-dimensional gas dynamics benchmark problems.

Bicompact Schemes for an Inhomogeneous Linear Transport Equation

The bicompact finite-difference schemes constructed for a homogeneous linear transport equation for the case of the inhomogeneous transport equation are generalized. The equation describes the transport of particles or radiation in media. Using the method of lines, the bicompact scheme is constructed for the initial unknown function and the complementary unknown mesh function defined as the integral average of the initial function with respect to space cells. The comparison of the calculation results of the proposed method and the conservative-characteristic method is carried out. The latter can be assigned to the class of bicompact finite-difference schemes; however, this method is based on the idea of the redistribution of incoming fluxes from illuminated edges to unilluminated edges.

Monotonization of a highly accurate bicompact scheme for a stationary multidimensional transport equation

A variant of a hybrid scheme for solving the nonhomogeneous stationary transport equation is constructed. A bicompact scheme of the fourth order approximation over all space variables and the first order approximation scheme from a set of short characteristic methods with interpolation over illuminated faces are chosen as a base. It is shown that the chosen first order approximation scheme is a scheme with minimal dissipation. A monotonic scheme is constructed by a continuous and homogeneous procedure in all the mesh cells by keeping the fourth approximation order in domains where the solution is smooth and maintaining a high level of accuracy in the domain of the discontinuity. The logical simplicity and homogeneity of the suggested algorithm make this method well fitted for supercomputer calculations.

A sixth-order bicompact scheme with spectral-like resolution for hyperbolic equations

For the numerical solution of nonstationary quasilinear hyperbolic equations, a family of symmetric semidiscrete bicompact schemes based on collocation polynomials is constructed in the one- and multidimensional cases. A dispersion analysis of a semidiscrete bicompact scheme of six-order accuracy in space is performed. It is proved that the dispersion properties of the scheme are preserved on highly nonuniform spatial grids. It is shown that the phase error of the sixth-order bicompact scheme does not exceed 0.2% in the entire range of dimensionless wave numbers. A numerical example is presented that demonstrates the ability of the bicompact scheme to adequately simulate wave propagation on coarse grids at long times.

Iterative approximate factorization for difference operators of high-order bicompact schemes for multidimensional nonhomogeneous hyperbolic systems

An iterative method for solving equations of multidimensional bicompact schemes based on an approximate factorization of their difference operators is proposed for the first time. Its algorithm is described as applied to a system of two-dimensional nonhomogeneous quasilinear hyperbolic equations. The convergence of the iterative method is proved in the case of the two-dimensional homogeneous linear advection equation. The performance of the method is demonstrated on two numerical examples. It is shown that the method preserves a high (greater than the second) order of accuracy in time and performs 3–4 times faster than Newton’s method. Moreover, the method can be efficiently parallelized.