Third-order Difference Scheme Research Articles

TVD-модификация метода Годунова 3-го порядка

A study of the monotonicity properties of a new modification of the Godunov method, which has a third order of approximation in space and time, is presented. The difference scheme of the method uses a simultaneous discretization of conservation laws in space and time without of Runge — Kutta stages. This method is a development of the second order Godunov method through the connection of two additional fluxes correction procedures. The first procedure increases the order of approximation for linear systems of equations. The second procedure uses the finite differences of the Jacobi matrix of the system of equations and eliminates the second-order error that occurs due to the nonlinearity of the equations. The TVD property of the difference scheme is strictly proved in relation to the linear scalar transfer equation when using an generalized adaptive limiter of central differences. New modifications of limiters are proposed that significantly improve the accuracy of the solution in the vicinity of discontinuities and local extremes. New limiters are obtained from the known ones using a simple function of local smooth deformation. An economical version of the third-order Godunov method for gas dynamics equations with quadratic reconstruction of parameters in primary variables is presented. The use of primary variables for reconstruction significantly reduces the FPU time during calculations. The advantages of the third-order method in terms of the accuracy of the solution in the vicinity of contact discontinuities and the boundaries of the rarefaction wave are demonstrated on standard tests. The proposed approach to constructing third-order difference schemes can be used for inhomogeneous and two-dimensional hyperbolic systems of nonlinear equations.

Hermitian Characteristic Scheme for a Linear Inhomogeneous Transfer Equation

In this paper, we design an interpolation-characteristic scheme for the numerical solution of the inhomogeneous transfer equation. The scheme is based on the Hermitian interpolation to reconstruct the value of an unknown function at the point of intersection of the backward characteristic with the cell faces. The Hermitian interpolation to reconstruct the function values ​​uses both the nodal values ​​of the desired function and its derivative. Unlike previous studies also based on the Hermitian interpolation, not only the differential continuation of the transfer equation but also the relationship between the integral averaged values, nodal values, ​​and derivatives according to the Euler-Maclaurin formula is used to transfer information about the derivatives to the next layer. The third-order difference scheme is shown to converge for smooth solutions. The dissipative and dispersive properties of the scheme are considered using numerical examples of solutions with decreasing smoothness.

The Third-Order Approximation Godunov Method for the Equations of Gas Dynamics

The paper suggests a new modification of Godunov difference method with the 3rd order approximation in space and time for hyperbolic systems of conservation laws. The diёerence scheme uses the simultaneous discretization of the equations in space and time without of Runge — Kutta stages. An exact or approximate solution of Riemann problem is applied to calculate numerical fluxes between cells. Before the time step, corrections to the arguments of the Riemann problem providing third-order approximations for linear systems are calculated. After the time step, the numerical solution correction procedure is applied to eliminate the second-order error arising from the nonlinearity of the equations. The paper presents the results of experimental numerical verification of the method approximation order on the exact smooth solution inside the fan of the expansion wave. The test results completely confirm the third order of the presented method. The proposed approach of constructing third-order difference schemes can be used for inhomogeneous and two-dimensional hyperbolic systems of nonlinear equations.

Design of High-Order Difference Scheme and Analysis of Solution Characteristics—Part II: A Kind of Third-Order Difference Scheme and New Scheme Design Theory

In this Part II, on the basis of the general style design of second-order difference scheme and the analysis of the absolutely stable scheme proposed in Part I, the companion article, the general design method of any high-order difference scheme is proposed. Based on this method, a new kind of third-order difference scheme including 17 different variants is constructed, which uses the same grid points as existing second-order difference schemes but is different from them in that the grids are chosen symmetrically from two sides of the interface. Because they have the same matrix style created by the same grid plots of the discretization equation, these third-order schemes require the same CPU time and memory as the second-order schemes; however, this kind of symmetrical third-order difference scheme will keep the consistency between the false diffusion and the stability, and the stability of the scheme is better than that of the existing biased second-order scheme. Further research shows that under the conditions of matrix style and computer memory, the scheme constituted by symmetrically numbered grids from two sides of the interface with odd order of accuracy can maintain consistency between numerical accuracy and stability better than any kind of scheme designed according to the “upwind” idea. Based on this understanding, a new scheme design theory called symmetric and odd-order accuracy scheme design theory is proposed.

Numerical modelling of viscoelastic liquids using a finite-volume method

A new staggered-grid, finite-volume method for the numerical simulation of isothermal viscoelastic liquids is presented. The main features of this method are the use of a primitive variable formulation, the location of the velocities, pressures and stresses on different staggered grids and the use of a third-order difference scheme for the discretization of the constitutive equations. The method is applied to a sudden-expansion, viscoelastic-flow problem for a range of Weissenberg numbers. For one case the mesh has been refined to demonstrate that the method is robust and viscoelastic effects are not filtered out as the mesh size decreases. All the computations have been performed on a PC equipped with a floating-point coprocessor. Although the method has been defined for, and applied to, two-dimensional problems, the technique is readily extendable to three dimensions without excessive cost.

A study of upstream‐weighted high‐order differencing for approximation to flow convection

AbstractThis paper is concerned with a number of upstream‐weighted second‐ and third‐order difference schemes. Also considered are the conventional upwind and central difference schemes for comparison. It commences with a general difference equation which unifies all the given first‐, second‐ and third‐order schemes. The various schemes are evaluated through the use of the general equation. The unboundedness and accuracy of the solutions by the difference schemes are assessed via various analyses: examination of the coefficients of the difference equation, Taylor series truncation error analysis, study of the upstream connection to numerical diffusion, single‐cell analysis. Finally, the difference schemes are tested on one‐ and two‐dimensional model problems. It is shown that the high‐order schemes suffer less from the problem of numerical diffusion than the first‐order upwind difference scheme. However, unboundedness cannot be avoided in the solutions by these schemes. Among them the linear upwind difference scheme presents the best compromise between numerical diffusion and solution unboundedness.