Numerical solutions of 2D Navier-Stokes equations based on generalized harmonic polynomial cell method with non-uniform grid

It is essential for a Navier-Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present numerical solutions of the 2D Navier-Stokes equations using the fourth-order generalized harmonic polynomial cell (GHPC) method as the Poisson equation solver. Particular focus is on the local and global accuracy of the GHPC method on non-uniform grids. Our study reveals that the GHPC method enables use of more stretched grids than the original HPC method. Compared with a second-order central finite difference method (FDM), global accuracy analysis also demonstrates the advantage of applying the GHPC method on stretched non-uniform grids. An immersed boundary method is used to deal with general geometries involving the fluid-structure-interaction problems. The Taylor-Green vortex and flow around a smooth circular cylinder and square are studied for the purpose of verification and validation. Good agreement with reference results in the literature confirms the accuracy and efficiency of the new 2D Navier-Stokes equation solver based on the present immersed-boundary GHPC method utilizing non-uniform grids. The present Navier-Stokes equations solver uses second-order FDM for the discretization of the diffusion and advection terms, which may be replaced by other higher-order schemes to further improve the accuracy.

Symmetry-Breaking in a Porous Cavity with Moving Side Walls

In this paper, a numerical investigation of the steady laminar mixed convection flow in a porous square enclosure has been considered. This structure represents a practical system such as an external through flow of cooled-air an electronic device from its moving sides. The heating was supplied by an internal volumetric source with an uniform distribution at the middle part of its bottom, while the other walls were assumed thermally insulated. Moreover, the momentum transfer in the porous substrate was numerically investigated using the Darcy-Brinkman-Forchheimer law. The governing equations of the posed problem have been solved by applying the finite difference technique on non-uniform grids. For all simulations, the Reynolds number and the porosity have been fixed respectively to Re=100 and φ=0.9. Darcy’s value was varied in the range from 0.001 to 0.1. The results detected the existence of a radical change in the contour patterns for Richardson number equal to 11.76 and 11.77 with fixed Da=0.1. This behavior signified that the fluid is fully convected for higher Darcy number.

Theoretical Analysis of Fully Conservative Difference Schemes with Adaptive Viscosity

For the equations of gas dynamics in Eulerian variables, a family of two-layer in time completely conservative difference schemes with space-profiled time weights is constructed. Considerable attention is paid to the methods of constructing regularized flows of mass, momentum, and internal energy that do not violate the properties of complete conservatism of difference schemes of this class, to the analysis of their amplitudes and the possibility of their use on non-uniform grids. Effective preservation of the balance of internal energy in this type of divergent difference schemes is ensured by the absence of constantly operating sources of difference origin that produce "computational"entropy (including those based on singular features of the solution). The developed schemes can be easily generalized in order to calculate high-temperature flows in media that are nonequilibrium in temperature (for example, in a plasma with a difference in the temperatures of the electronic and ionic components), when, with the set of variables necessary for describing the flow, it is not enough to equalize the total energy balance.

Numerical Results of Crank-Nicolson and Implicit Schemes to Laplace Equation with Uniform and Non-Uniform Grids

In this paper, we investigate the numerical results between Implicit and Crank-Nicolson method for Laplace equation. Based on the numerical results obtained, we get the conclusion that the absolute error of Crank-Nicolson method is smaller than the absolute error of Implicit method for uniform and non-uniform grids which both refer to the analytical solution of Laplace equation obtained by separable variable method.Keywords: Crank-Nicolson; Implicit; Laplace equation; separable variable method; uniform and non-uniform grids. AbstrakDalam makalah ini, kami menyelidiki hasil numerik antara etode Implisit dan Crank-Nicolson untuk persamaan Laplace. Berdasarkan hasil numerik yang diperoleh, kita mendapatkan kesimpulan bahwa kesalahan absolut metode Crank-Nicolson lebih kecil daripada kesalahan absolut metode Implisit untuk grid seragam dan tak-seragam yang keduanya mengacu pada solusi analitik persamaan Laplace yang diperoleh dengan metode separable.Kata kunci: Crank-Nicolson; Implisit; persamaan Laplace; metode variable terpisah; grid seragam dan tak-seragam.

A three-dimensional atmospheric dispersion model for Mars

Atmospheric local-to-regional dispersion models are widely used on Earth to predict and study the effects of chemical species emitted into the atmosphere and to contextualize sparse data acquired at particular locations and/or times. However, to date, no local-to-regional dispersion models for Mars have been developed; only mesoscale/microscale meteorological models have some dispersion and chemical capabilities, but they do not offer the versatility of a dedicated atmospheric dispersion model when studying the dispersion of chemical species in the atmosphere, as it is performed on Earth. Here, a new three-dimensional local-to-regional-scale Eulerian atmospheric dispersion model for Mars (DISVERMAR) that can simulate emissions to the Martian atmosphere from particular locations or regions including chemical loss and predefined deposition rates, is presented. The model can deal with topography and non-uniform grids. As a case study, the model is applied to the simulation of methane spikes as detected by NASA’s Mars Science Laboratory (MSL); this choice is made given the strong interest in and controversy regarding the detection and variability of this chemical species on Mars.

Improved wave dispersion properties in 1D and 2D bond-based peridynamic media

In this study, a novel method for improving the simulation of wave propagation in Peridynamic (PD) media is investigated. Initially, the dispersion properties of the nonlocal Bond-Based Peridynamic model are computed for 1-D and 2-D uniform grids. The optimization problem, developed through inverse analysis, is set up by comparing exact and numerical dispersion and minimizing the error. Various weighted residual techniques, i.e., point collocation, sub-domain collocation, least square approximation and the Galerkin method, are adopted and the modification of the wave dispersion is then proposed. It is found that the proposed methods are able to significantly improve the description of wave dispersion phenomena in both 1-D and 2-D PD models.

The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation

In this paper, we study a loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference schemes of Crank-Nicolson type of higher order is developed for approximating the stated problem on uniform grids with the orders of accuracy O ( h 4 + τ 2 − α ) $\mathcal{O}(h^4+\tau^{2-\alpha})$ and O ( h 4 + τ 2 ) $\mathcal{O}(h^4+\tau^{2})$ . A priori estimates are obtained for solutions of differential and difference equations. Stability of the suggested schemes and also their convergence with the rate equal to the order of the approximation error are proved. Proposed theoretical calculations are illustrated by numerical experiments on test problems.

Analogs of S.N. Bernstein and V.I. Smirnov inequalities for harmonic polynomials

Гармонические отображения и, в частности, гармонические полиномы находят приложения во многих задачах прикладной математики, математической физики, механики и электротехники. Это связано с ключевой ролью, которую гармонические функции играют в краевых задачах математической физики. Гармонические полиномы используются при описании плоских гармонических векторных полей в гидродинамике, в теории жидких кристаллов, в теории плоского потенциала. Оценки гармонических полиномов и их производных применяются при разработке неравномерных сеток и триангуляций во многих вычислительных схемах и математическом моделировании. В середине двадцатого столетия советскими математиками С.Н. Бернштейном и В.И. Смирновым были доказаны результаты из области дифференциальных неравенств, связывающих многочлены $P(z)=a_n z^n+ a_{n-1} z^{n-1}+ \dots a_1 z+ a_0$ в комплексной плоскости $\mathbb{C}$ и их производные $P'(z)$. Данная тематика сохраняет актуальность, о чем свидетельствует большое число посвященных ей новых публикаций российских и зарубежных математиков. В настоящей работе доказаны результаты, обобщающие неравенства С.Н. Бернштейна и В.И. Смирнова на случай гармонических многочленов $F=H+\overline G,$ где $H, G$ - аналитические многочлены. В частности получены условия типа мажорирующих неравенств на единичной окружности, позволяющие связать производные аналитических и антианалитических частей гармонических многочленов, все нули которых расположены в единичном круге. Доказательства основных результатов получены с помощью топологического аналога известного в теории функций принципа аргумента, позволяющего свести некоторые задачи теории гармонических многочленов к аналитическому случаю. Из полученных результатов следуют классические неравенства Смирнова и Бернштейна в случае аналитических многочленов. Доказанные теоремы проиллюстрированы примером, демонстрирующим точность сформулированных нами условий и оценок. Harmonic mapings and, in particular, harmonic polynomials find applications in many problems of mathematics, mathematical physics, mechanics and electrical engineering. This is due to the key role that harmonic functions play in boundary value problems of mathematical physics. Harmonic polynomials are used to describe plane harmonic vector fields in hydrodynamics, in the theory of liquid crystals, in the theory of plane potential. Estimates of harmonic polynomials and their derivatives are used in the development of non-uniform grids and triangulations in many computational schemes. In the middle of the twentieth century, Soviet mathematicians S.N. Bernstein and V.I. Smirnov proved results several differential inequalities connecting the polynomials $P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots a_1 z + a_0$ in the complex plane $\mathbb{C}$ and their derivatives. This topic remains important, as evidenced by the large number of new publications of Russian and foreign mathematicians. In this paper, we proved results that generalize the inequalities of S.N. Bernstein and V.I. Smirnov for the case of harmonic polynomials $F = H + \overline G,$ where $H, G$ are analytic polynomials. In particular, conditions of the type of majorizing inequalities on the unit circle are obtained, which make it possible to estimate the derivatives of the analytic and antianalytic parts of harmonic polynomials, all of whose zeros are located in the unit disk. The proofs of the main results are obtained using a topological analogue of the principle of the argument known in the theory of functions, which makes it possible to reduce some problems of the theory of harmonic polynomials to the analytic case. The classical inequalities of Smirnov and Bernstein in the case of analytic polynomials follow from the results of current paper. The proved theorems are illustrated by an example that demonstrates the accuracy of the conditions and estimates formulated by us.