Solving the hourglass instability problem using rare mesh variation-difference schemes

The hourglass instability effect is characteristic of the Wilkins explicit difference scheme or similar schemes based on two-dimensional 4-node or three-dimensional 8-node finite elements with one integration point in the element. The hourglass effect is absent in schemes with cells in the form of simplexes (triangles in two-dimensional case, tetrahedrons in three-dimensional case). But they have another well-known drawback - slow convergence. One of the authors proposed a rare mesh scheme, in which elements in the form of a tetrahedron are located one at a time in the centers of the cells of a hexahedral grid. This scheme showed the absence “hourglass” effect and other drawbacks with high efficiency. This approach was further developed for solving 2D and 3D problems.

A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation

<abstract><p>The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.</p></abstract>

High order modified differential equation of the Beam–Warming method, II. The dissipative features

This paper is a continuation of [38]. The analysis of the modified partial differential equation (MDE) of the constant-wind-speed linear advection equation explicit difference scheme up to the eighth-order derivatives is presented. In this paper the authors focus on the dissipative features of the Beam–Warming scheme. The modified partial differential equation is presented in the so-called Π-form of the first differential approximation. The most important part of this form includes the coefficients μ (p) at the space derivatives. Analysis of these coefficients provides indications of the nature of the dissipative errors. A fragment of the stencil for determining the modified differential equation for the Beam–Warming scheme is included. The derived and presented coefficients μ (p) as well as the analysis of the dissipative features of this scheme on the basis of these coefficients have not been published so far.

High order modified differential equation of the Beam–Warming method, I. The dispersive features

The analysis of the modified partial differential equation (MDE) of the constant wind speed advection equation explicit difference scheme up to the eighth order with respect to both space and time derivatives is presented. So far, in majority of publications this modified equation has been derived mainly as a fourth-order equation. The MDE is presented in the so-called Π-form of the first differential approximation. This form includes only the space derivatives of higher order p and their coefficients μ(p). Analysis of these coefficients provides indications of the nature of the dissipative and dispersive errors. A fragment of the stencil for determining the modified differential equation up to the eighth-order MDE for the second-order Beam–Warming scheme is included. The derived coefficients μ(p) as well as the analysis of the phase shift errors, the phase and group velocities and dispersive features on the basis of these coefficients have not been published so far. The dissipative features of this method we present in [33].

Stability Analysis of the Explicit Difference Scheme for Richards Equation

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.

Simulation of 3D Wave Fields in Inhomogeneous Domain with Complex Topography Using the Lebedev Scheme

Numerical simulation is widely used in the study of wave fields in various media. One of the methods is to divide the domain of interest into elementary volumes and build a finite-difference scheme for numerical implementation. The work assumes that the domain can have a significant curvature of the surface, therefore, the technology of generating a mesh of curved cubes is used. This mesh provides good consistency between the discrete and physical models of the domain. A parallel algorithm is proposed for the numerical solution of a 3D linear system of elasticity theory, expressed via displacement velocities and stresses, using a curvilinear mesh and an explicit difference scheme based on the Lebedev scheme. The simulation results are presented. The calculations were carried out using the resources of the SSCC SB RAS.

CALCULATION OF THERMODYNAMIC CHARACTERISTICSOF THE ASCENDING SWIRLING FLOW WITH FIVE SOURCES OF HEATING

Calculations of the thermodynamic characteristics of air flows were carried out when several local sources were heated by the underlying surface. The basic mathematical model is the complete system of Navier-Stokes equations with constant coefficients of viscosity and thermal conductivity when gravity and Coriolis are taken into account. Numerical experiments were carried out using an explicit difference scheme with an appropriate choice of the initial and boundary conditions. The varying local pressure differences found during the calculations lead to the corresponding gas flows both in horizontal and vertical planes.

Difference Approximation of Stochastic Elastic Equation Driven by Infinite Dimensional Noise

An explicit difference scheme is described, analyzed and tested for numerically approximating stochastic elastic equation driven by infinite dimensional noise. The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series. Error analysis of the numerical method yields estimate of convergence rate. The rate of convergence is demonstrated with numerical experiments.