Bicompact Schemes Research Articles

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.

On spectral-like resolution properties of fourth-order accurate symmetric bicompact schemes

A dispersion analysis is conducted for bicompact schemes of fourth-order accuracy in space, namely, for a semidiscrete scheme and a second-order accurate scheme in time. It is shown that their numerical group velocity is positive for all dimensionless wavenumbers. It is proved that the dispersion properties of the bicompact schemes are preserved on highly nonuniform meshes. A comparison reveals that the fourth-order bicompact schemes have a higher spectral resolution than not only other same-order compact schemes, but also some sixth-order ones. Two numerical examples are presented that demonstrate the ability of the bicompact schemes to adequately simulate wave propagation on highly nonuniform meshes over long time intervals.

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.

Bicompact schemes for solving a steady-state transport equation by the quasi-diffusion method

Accurate difference schemes (of up to the fourth order of accuracy) are constructed for the numerical solutions of a neutron-transport equation and a system of quasi-diffusion equations (a low order consequences of a transport equation) used to accelerate iterations on scattering. The considered difference schemes are based on the common principles of the compact (in the context of a single cell) approximation. This enables one to make accurate resolution for contact discontinuities in the medium. The fourth order of approximation on a minimum two-point stencil is attained by widening the list of unknowns and including on it (apart from the nodal values of an unknown function) additional unknowns. As these unknowns, an integral cell-average value or a value at a half-integer node can be taken. To connect these values, Simpson quadrature formulas are used. Equations for determining additional unknowns are constructed by using the Euler–Maclaurin formulas. Computations for several onedimensional test problems have been performed; here, the high actual accuracy of the constructed difference schemes is demonstrated. The schemes are naturally generalized to the two- and three-dimensional cases. Due to their high accuracy, monotonicity, efficiency, and compactness, the proposed schemes are very attractive for engineering computations (computations of nuclear reactors, etc.).

High-order accurate bicompact schemes for solving the multidimensional inhomogeneous transport equation and their efficient parallel implementation

The method of lines is used to obtain semidiscrete equations for a bicompact scheme in operator form for the inhomogeneous linear transport equation in two and three dimensions. In each spatial direction, the scheme has a two-point stencil, on which the spatial derivatives are approximated to fourth-order accuracy due to expanding the list of unknown grid functions. This order of accuracy is preserved on an arbitrary nonuniform grid. The equations of the method of lines are integrated in time using diagonally implicit multistage Runge–Kutta methods of the third up fifth orders of accuracy. Test computations on refined meshes are presented. It is shown that the high-order accurate bicompact schemes can be efficiently parallelized on multicore and multiprocessor computers.

On exact dimensional splitting for a multidimensional scalar quasilinear hyperbolic conservation law

A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time-stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero.

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.

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

A hybrid scheme is proposed for solving the nonstationary inhomogeneous transport equation. The hybridization procedure is based on two baseline schemes: (1) a bicompact one that is fourth-order accurate in all space variables and third-order accurate in time and (2) a monotone first-order accurate scheme from the family of short characteristic methods with interpolation over illuminated faces. It is shown that the first-order accurate scheme has minimal dissipation, so it is called optimal. The solution of the hybrid scheme depends locally on the solutions of the baseline schemes at each node of the space-time grid. A monotonization procedure is constructed continuously and uniformly in all mesh cells so as to keep fourth-order accuracy in space and third-order accuracy in time in domains where the solution is smooth, while maintaining a high level of accuracy in domains of discontinuous solution. Due to its logical simplicity and uniformity, the algorithm is well suited for supercomputer simulation.

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.

On bicompact grid-characteristic schemes for the linear advection equation

A set of grid-characteristic schemes for the linear advection equation is considered. Depending on the behavior of the solution, hybrid compact difference schemes of second–third order accuracy are proposed as based on interpolation polynomials. The schemes produce monotone solutions and only slightly smear discontinuities.

Hybrid running schemes with upwind and bicompact symmetric differencing for hyperbolic equations

New hybrid difference schemes are proposed for computing discontinuous solutions of hyperbolic equations. Involved in these schemes, a bicompact scheme that is third-order accurate in time and fourth-order accurate in space is monotonized using several partner schemes, namely, a first-order accurate explicit upwind scheme and two bicompact schemes of second and fourth orders of accuracy in space, both of the first order of accuracy in time. Their total domain of monotonicity covers all Courant numbers. An algorithm for automatically choosing the most suitable partner scheme is constructed. The mechanism of switching between high- and low-order accurate schemes is rigorously substantiated. All the schemes used can be efficiently implemented by applying the running calculation method. The hybrid schemes proposed have been tested on a model two-dimensional explosion problem in an ideal gas.

Uniqueness of a high-order accurate bicompact scheme for quasilinear hyperbolic equations

The possibility of constructing new third- and fourth-order accurate differential-difference bicompact schemes is explored. The schemes are constructed for the one-dimensional quasilinear advection equation on a symmetric three-point spatial stencil. It is proved that this family of schemes consists of a single fourth-order accurate bicompact scheme. The result is extended to the case of an asymmetric three-point stencil.

Bicompact schemes for an inhomogeneous linear transport equation in the case of a large optical depth

B.V. Rogov’s compact difference schemes are considered, which are constructed for an inhomogeneous linear transport equation inherent in the description of the transport problems of radiation or particles in a medium. The transition from a multigroup description of the energy dependence of the transport equation solution to the Lebesgue averaging method implies a strong expansion in the range of the absorption coefficient variation, especially in the direction of its increase. This paper proposes a new method for the solution of the monotonization of problems with a large optical depth, which significantly improves the accuracy of the solution in the case of its nondifferentiability, with an accuracy close to that of conservative characteristic schemes.

Bicompact scheme for the multidimensional stationary linear transport equation

A fourth-order accurate (in space) bicompact scheme is proposed for solving the inhomogeneous stationary transport equation in two dimensions. The scheme is based on a minimal stencil consisting of two nodes in each dimension and is obtained as a stationary limit of bicompact schemes produced by the method of lines for the nonstationary transport equation. The set of unknowns for each two-dimensional cell consists of the node values of the solution function and its integrals over cell edges and the entire cell. A closed system of linear equations is obtained for all desired variables in each cell. This system is solved using the running calculation method, which reveals the characteristic properties of the transport equation without explicitly using characteristics. The numerical results are compared with the solution produced by a conservative-characteristic method applied to a similar set of variables. The advantages of the bicompact schemes are demonstrated.

Bicompact schemes for an inhomogeneous linear transport equation

The bicompact finitedifference schemes co nstructed 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 func� tion 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 conservativecharacteristic method is carried out. The latter can be assigned to the class of bicompact finitedifference schemes; however, this method is based on the idea of the redistribution of incoming fluxes from illuminated edges to unillu� minated edges.

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.

Monotone bicompact scheme for quasilinear hyperbolic equations

Hyperbolic equations and systems of equationscomprise a considerable part of mathematical modelsused in various applications [1]. Among them, someproblems are related to finding weak solutions, whichare described by piecewise smooth functions. Highorder accurate shockcapturing schemes are widelyused for the numerical solution of nonlinear problems.Important properties to be possessed by such schemesare monotonicity [1] and conservation [2]. For thisreason, the design of monotone and conservative difference schemes has been addressed in numerous publications [1, 2].For hyperbolic quasilinear equations and systemsof equations, bicompact difference schemes on a twopoint stencil were proposed in [3]. They have fourthorder accuracy in space and are absolutely stable, conservative, and monotone for local Courant numbersκ≥ [4]. Moreover, they are efficient and can besolved using by the running calculation method.In this paper, a new highorder accurate monotonebicompact scheme is proposed for quasilinear hyperbolic equations. Like the schemes in [3], it is absolutely stable, conservative, and efficient but, in contrast to the former, has the following additional advantages. First, the new scheme remains monotone forsmaller Courant numbers than that in [3]. Second,when applied to stiff problems, the scheme is morerobust than that in [3]. An

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.

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 bicompact schemes for a linear advection equation

approximated by fourthorder compact differences, and an extended system is used. In contrast to (9), the original equation for the unknown function was sup� plemented not with an equation for its spatial deriva� tives but rather with an equation for its primitive. An advantage of this approach is obvious in the case of shockcapturing difference schemes as applied to the computation of discontinuous solutions, since the smoothness of a function's primitive is higher by one than that of the function itself. In (10) bicompact dif� ference schemes for a linear advection equation were constructed using various timestepping schemes as applied to evolution systems of differential equations obtained by the method of lines. However, the proper� ties of these schemes were examined only for smooth solutions. First, we show that the bicompact homogeneous scheme of (10), which is firstorder accurate in time and fourthorder accurate in space, as applied to a lin� ear advection equation is monotone. Second, this baseline scheme is used to construct a monotonized highorder accurate scheme in time. Both schemes were solved explicitly using the running calculation method. The numerical results produced by these schemes were compared with wellknown highorder accurate essentially nonoscillatory schemes (11, 12). The comparison showed the superiority of the schemes proposed in this paper.