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With the advent of technology, it has become possible to perform direct numerical simulations and the demand for high accuracy computing is increasing. Numerical simulations play an important part in understanding physics of the flow and instability mechanism in flows. For high accuracy, numerical schemes must be chosen that satisfy the physical dispersion relation, should not amplify or attenuate the solution and resolve all possible length and time scales. In the present paper, spectral stability analysis of linear convection equation is performed using first order forward difference (FD1) method and fourth order Runge Kutta (RK4) method, consisting of four stages, for time discretization and a second order central difference (CD2) method for evaluating spatial derivative. The results show that the presence of numerical instability for FD1 method is independent of the CFL number, consistent with the stability analysis which showed FD1 method to be unconditionally unstable. However, for RK4 method, the solution is found to be neutrally stable only for a particular range of CFL number, even stable solution introduced error by attenuating the computed or analytical solution.

With the advent of technology, it has become possible to perform direct numerical simulations and the demand for high accuracy computing is increasing. Numerical simulations play an important part in understanding physics of the flow and instability mechanism in flows. For high accuracy, numerical schemes must be chosen that satisfy the physical dispersion relation, should not amplify or attenuate the solution and resolve all possible length and time scales. In the present paper, spectral stability analysis of linear convection equation is performed using first order forward difference (FD1) method and fourth order Runge Kutta (RK4) method, consisting of four stages, for time discretization and a second order central difference (CD2) method for evaluating spatial derivative. The results show that the presence of numerical instability for FD1 method is independent of the CFL number, consistent with the stability analysis which showed FD1 method to be unconditionally unstable. However, for RK4 method, the solution is found to be neutrally stable only for a particular range of CFL number, even stable solution introduced error by attenuating the computed or analytical solution. [ABSTRACT FROM AUTHOR]

We study the random flow, through a thin cylindrical tube, of a physical quantity of random density, in the presence of random sinks and sources. We model convection in terms of the expectations of the flux and density and solve the initial value problem for the resulting convection equation. We propose a difference scheme for the convection equation, that is both stable and satisfies the Courant–Friedrichs–Lewy test, and estimate the difference between the exact and approximate solutions.

We study the random flow, through a thin cylindrical tube, of a physical quantity of random density, in the presence of random sinks and sources. We model convection in terms of the expectations of the flux and density and solve the initial value problem for the resulting convection equation. We propose a difference scheme for the convection equation, that is both stable and satisfies the Courant–Friedrichs–Lewy test, and estimate the difference between the exact and approximate solutions. [ABSTRACT FROM AUTHOR]

This study considers discontinuous Galerkin schemes with a Legendre polynomial basis of degree K = 0, ..., 5. The schemes built for the convection equation are studied in terms of the amplitude and dispersion error. The relation between wave number and cell size is determined to provide the desired accuracy of the solution in the task with a harmonic wave. The computations for an acoustic wave are performed with the program for full Euler equations. Despite the different problem statement, the analytical estimates prove their applicability for practical use. The other test considers inviscid a vortex flow. The practical estimate of the mesh density is based on the solution of the model vortex flow with viscosity. The relationship between the order of the scheme and mesh density is pointed out for all the test cases. The results of this paper can be used to build meshes for tasks intended to be solved with the discontinuous Galerkin approach. [ABSTRACT FROM AUTHOR]

An improved r -factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity. [ABSTRACT FROM AUTHOR]

In this research paper, a weighed DG finite element method is proposed for solving convection equations with an easy execution and analysis. The key aim of this method is to design an error estimation for space and time of a discontinuous approximation on general finite element meshes. The efficiency of the parameter θ in the order of convergence of the solutions is also exposed. Some numerical examples were tested that demonstrated the strength and flexibility of the method.

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as ε-1, ε « 1 being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of ε by using some two- scale convergence tools. Following Frénod & Sonnendrücker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency ε-1 » 1. In this abstract framework, we derive an expansion with respect to the small parameter ε and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams. [ABSTRACT FROM AUTHOR]

The diagnostic model DIAMOD has been developed in Parts l and II of this study. Its core is the convection equation for the vertical sub-gridscale flux of moist enthalpy, called total convective heat flux h. The convection equation is forced by the analyzed gridscale total heat budget. It has been implied in Part II that the size of h is controlled by the mean vertical derivative (the slope) of the budget, mainly in cases of strong convection. The slope of the budget, and consequently h, is quite sensitive to the numerical representation of the convection equation. The original implementation of DIAMOD in pressure coordinates (p-DIAMOD) suffers from the downward extrapolation of analysis data to standard pressure levels over high orography. This generates spurious values of the gridscale budget below the earth's surface which heavily affects the slope of the budget. The insufficient vertical resolution in the stratosphere also has a strong impact. Both problems are eliminated in the present implementation (η-DIAMOD) which is based on terrain following hybrid coordinates and discretized like the ECMWF forecast model. DIAMOD further yields an objective error estimate, the imbalance. While the vertical mean of the imbalance in the convection equation is a diagnosed quantity, its vertical profile has to be specified by an external error model. This profile is now chosen such that the impact upon h is minimized. The results obtained with the two model versions are compared for a rainstorm situation and contrasted with Yanai-type diagnostic studies. The h-profiles from η-DIAMOD in the free atmosphere are smaller than those from p-DIAMOD. However, they are better in accord with the meteorological situation.

An oscillation-free high order scheme is presented for convection discretization by using the normalized-variable formulation in the finite volume framework. It adopts the cubic upwind interpolation scheme as the basic scheme so as to obtain high order accuracy in smooth solution domain. In order to avoid unphysical oscillations of numerical solutions, the present scheme is designed on the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical results of several linear and nonlinear convection equations with smooth or discontinuous initial distributions demonstrate the present scheme possesses second-order accuracy, good robustness and high resolution. [ABSTRACT FROM AUTHOR]

Finite difference approximations for the convection equation are developed, which exhibit enhanced stability limits for explicit Runge–Kutta integration. Stability limits are increased by adding artificial dissipation terms, which are optimized to yield greatest stable time steps. For the artificial dissipation terms, symmetric finite difference approximations of even-order derivatives are used with differencing stencils equal to the convective stencils. The spatial discretization inclusive of the added dissipation term is shown to be consistent with a first derivative. The formal order of accuracy in space is decreased by one order, while the order of time integration is not affected. As a result, the time step limits of originally stable Runge–Kutta integration is increased, for some combinations of spatial discretization and time integration by a factor of two. Algorithms, which are unstable without damping are stabilized. The dispersion properties of the algorithms are not influenced by the proposed damping terms. Spectral analysis of the algorithms show very low dissipation error for dimensionless wave numbers k Δ x < 0.5. Stability conditions based on von Neumann stability analysis are given for the proposed schemes for explicit Runge–Kutta time integration of orders up to ten. [ABSTRACT FROM AUTHOR]

In this study, a column-leaching experiment with two flow velocities was carried out to evaluate the estimation of boron transport by using two equations. The first equation, by neglecting the pore scale dispersion in homogeneous profile, was convection equation, CE, and the second one was advection dispersion equation, ADE. The special purpose of this study was to estimate boron leaching in homogenous soil column and investigate the effects of pore scale dispersivity under steady-state water flow conditions. In this leaching experiment, high water flow velocity resulted in a greater dispersion coefficient D, especially in sandy clay loam soil, compared with that in clay loam (CL). The agreement between the determined and estimated values with CE and ADE was poor in the sandy clay loam soil. On the other hand, in CL both values were in positive agreement, especially at the first leaching. This could be explained as follows: ADE was sensitive to the estimation of easily soluble boron at first, and when easily soluble boron decreased with leaching, the model estimation began to deviate from the determined values. However, slow and faster water flow velocities did not create any difference for CL soil. [ABSTRACT FROM AUTHOR]

A novel, comprehensive, discrete, half-space analysis for the defect-correction method has been developed. This analysis plays the same role for nonelliptic-problem solvers as the full-space Fourier mode analysis plays for elliptic-problem solvers. Numerical simulations confirm the accuracy of the half-space analysis. The following important findings about the defect-correction method applied to the Fromm discretization of the two-dimensional convection equation are reported: 1. The initial convergence rate of the defect-correction method is principally a function of the relative accuracy of the operators involved in the defect-correction iterations. 2. The asymptotic convergence rate is about 0.5 per defect-correction iteration. 3. If the driver operator is first-order accurate, then the initial convergence rates may be slow. The number of iterations required to get into the asymptotic convergence regime or/and to converge the algebraic error below the discretization-error level can be proportional to h- 1/3. This h-dependent delay is a multidimensional phenomenon—it cannot be observed in one-dimensional problems, and it disappears in the case of close alignment between the grid and the convection equation characteristic. 4. If the driver operator is second-order accurate, the defect-correction solver demonstrates the asymptotic convergence rate from the very beginning. Only one defect-correction iteration is required to converge algebraic error substantially below the discretization-error level. [ABSTRACT FROM AUTHOR]

In this research paper, a weighed DG finite element method is proposed for solving convection equations with an easy execution and analysis. The key aim of this method is to design an error estimation for space and time of a discontinuous approximation on general finite element meshes. The efficiency of the parameter θ in the order of convergence of the solutions is also exposed. Some numerical examples were tested that demonstrated the strength and flexibility of the method. [ABSTRACT FROM AUTHOR]