Multi-moment advection scheme for Vlasov simulations

We present a new numerical scheme for solving the advection equation and its application to Vlasov simulations. The scheme treats not only point values of a profile but also its zeroth to second order piecewise moments as dependent variables, for better conservation of the information entropy. We have developed one-and two-dimensional schemes and show that they provide quite accurate solutions within reasonable usage of computational resources compared to other existing schemes. The two-dimensional scheme can accurately solve the solid body rotation problem of a gaussian profile for more than hundred rotation periods with little numerical diffusion. This is crucially important for Vlasov simulations of magnetized plasmas. Applications of the one- and two-dimensional schemes to electrostatic and electromagnetic Vlasov simulations are presented with some benchmark tests.     

Numerical modelling of unsteady convective–diffusive heat transfer with a control volume hybrid method

Presented in this paper is a numerical methodology for the solution of the parabolic governing partial differential equation that describes unsteady advection–diffusion heat transfer. The formulation presented here is shown to be free from the numerical oscillation commonly associated with advection–diffusion heat transfer regardless of the value of the Peclet number. The formulation involves the absorption of the advection term in the unsteady heat equation into the capacitance term. This process is achieved with the use of a control volume methodology applied to each nodal element on a finite-volume mesh. This is shown to ensure that spurious energy losses and gains are avoided and provides for consistency between temperature and energy change. This approach provides unconditional stability and it is shown that good accuracy is achievable with relatively large time-steps.     

Taylor–least-squares finite element for two-dimensional advection-dominated unsteady advection–diffusion problems

A new finite element method, the Taylor–least-squares, is proposed to approximate the advection-dominated unsteady advection–diffusion equation. The new scheme is a direct generalization of the Taylor–Galerkin and least-squares finite element methods. Higher-order spatial derivatives in the new formulation necessitate higher-degree polynomials. Hermite cubic shape functions are used. Extensive comparisons with other methods in one dimension proved that the new scheme is a step forward in modelling this difficult problem. The method offers straightforward generalizations to higher dimensions without losing the accuracy demonstrated in one dimension, i.e. the method preserves the important property of the Taylor–Galerkin scheme of being free of numerical crosswind diffusion. Several numerical experiments were made in two dimensions and excellent results were obtained from the representative experiments.     

An improvement on the positivity results for 2-stage explicit Runge-Kutta methods

In this paper, we investigate the positivity property for a class of 2-stage explicit Runge-Kutta (RK2) methods of order two when applied to the numerical solution of special nonlinear initial value problems (IVPs) for ordinary differential equations (ODEs). We also pay particular attention to monotonicity property. We obtain new results for positivity which are important in practical applications. We provide some numerical examples to illustrate our results.     

Air traverse time in grain bins

To maintain the quality of cereal grains during storage, it is necessary to keep the grain cool and free from insects, and typical methods for dealing with these problems are considered in this paper. In particular the insect population is controlled by fumigating the grain bed with carbon dioxide gas and the grain is cooled by forcing ambient air through the bed. In both problems, the equations which describe the physical processes contain a mixture of advection and diffusion or conduction terms. This paper explores the relationship between traverse time and heat and mass transfer and gains an insight into the grain storage processes that are controlled by forced convection. When heat and mass transport is dominated by the advection terms, the equations are simplified by changing variables from the (x,y) space coordinates to (ψ,τ), where ψ is the stream function for the problem and the traverse time τ at a point in the storage bin is the time taken for the air to travel to the point from the inlet duct. The conditions are described for the equations to be independent of ψ, with the main condition being that the derivatives of the metrics g₁₁, g₁₂ and g₂₂ with respect to ψ are small enough. If the equations are independent of ψ then the dependent variable (concentration or temperature) will be constant on lines of constant traverse time τ. This relationship between traverse time and the cooling or fumigation pattern can be used in the design of storage bins since it implies that the best outlet surface is a line of constant τ.     

An age structured population model in a spatially heterogeneous environment: Existence and uniqueness theory

This paper considers the existence and uniqueness of solutions of an age-structured population model in a spatially heterogeneous bounded environment. The model describes individual organism movement of diffusion and advection in the older life stage but assumes stationarity of individuals in the embryonic stage. The mathematical formulation is a nonlinear degenerate parabolic partial differential problem with non-locally integro-type initial-boundary conditions.     

Two-dimensional adaptive Eulerian-Lagrangian method for mass transport with spatial velocity distribution

A Eulerian-Lagrangian scheme is used to solve the two-dimensional advection-dispersion equation. Concentration and its partial differential operator are decomposed into advection and dispersion terms. Thus, advection is formally decoupled from dispersion and solved by continuous forward particle tracking. Dispersion is handled by implicit finite elements on a fixed Eulerian grid. Translation of steep gradients of concentration in advection-dominated flow regimes, is done without numerical distortion. Continuous spatial distribution of velocities are evaluated by using Galerkin's approach in conjunction with Darcy's law based on hydraulic input data from each element. The method was implemented on coarse FE grid with linear shape functions, demonstrating no over/under shooting and practically no numerical dispersion. Simulations, covering a wide range of Peclet numbers, yield high agreement with analytic and practical results.     

A higher-order eulerian scheme for coupled advection-diffusion transport

A new accurate high-order numerical method is presented for the coupled transport of a passive scalar (concentration) by advection and diffusion. Following the method of characteristics, the pure advection problem is first investigated. Interpolation of the concentration and its first derivative at the foot of the characteristic is carried out with a fifth-degree polynomial. The latter is constructed by using as information the concentration and its first and second derivatives at computational points on current time level t in Eulerian co-ordinates. The first derivative involved in the polynomial is transported by advection along the characteristic towards time level t + Δt in the same way as is the concentration itself. Second derivatives are obtained at the new time level t + Δt by solving a system of linear equations defined only by the concentrations and their derivatives at grid nodes, with the assumption that the third-order derivatives are continuous. The approximation of the method is of sixth order. The results are extended to coupled transport by advection and diffusion. Diffusion of the concentration takes place in parallel with advection along the characteristic. The applicability and precision of the method are demonstrated for the case of a Gaussian initial distribution of concentrations as well as for the case of a steep advancing concentration front. The results of the simulations are compared with analytical solutions and some existing methods.     

确定地下水污染强度的反问题方法

探讨了山东省淄博市张店区沣水南部区域地下水的硫酸污染问题 .根据观测井点的浓度数据 ,将硫酸污染强度的识别问题作为一个已知终值数据的源项反问题 .应用积分恒等式方法分析了未知源函数与已知数据间的数据相容性 ,进而基于最优化策略进行了数据反演 ,计算结果与实际估计值基本吻合