Conservative Upwind Schemes Research Articles

A high-accuracy Eulerian gyrokinetic solver for collisional plasmas

We describe a new approach to solve the electromagnetic gyrokinetic equations which is optimized for accurate treatment of multispecies Fokker–Planck collisions including both pitch-angle and energy diffusion. The new algorithm is spectral/pseudospectral in four of the five phase space dimensions, and in the fieldline direction a novel 5th-order conservative upwind scheme is used to permit high-accuracy electromagnetic simulation even in the limit of very high plasma β and vanishingly small perpendicular wavenumber, k⊥→0. To our knowledge, this is the first pseudospectral implementation of the collision operator in a gyrokinetic code. We show that the new solver agrees closely with GYRO in the limit of weak Lorentz collisions, but gives a significantly more realistic description of collisions at high collision frequency. The numerical methods are also designed to be efficient and scalable for multiscale simulations that treat ion-scale and electron–scale turbulence simultaneously.

Preserving bounded and conservative solutions of transport in one‐dimensional shallow‐water flow with upwind numerical schemes: Application to fertigation and solute transport in rivers

AbstractThis work intends to show that conservative upwind schemes based on a separate discretization of the scalar solute transport from the shallow‐water equations are unable to preserve uniform solute profiles in situations of one‐dimensional unsteady subcritical flow. However, the coupled discretization of the system is proved to lead to the correct solution in first‐order approximations. This work is also devoted to show that, when using a coupled discretization, a careful definition of the flux limiter function in second‐order TVD schemes is required in order to preserve uniform solute profiles. The work shows that, in cases of subcritical irregular flow, the coupled discretization is necessary but nevertheless not sufficient to ensure concentration distributions free from oscillations and a method to avoid these oscillations is proposed. Examples of steady and unsteady flows in test cases, river and irrigation are presented. Copyright © 2007 John Wiley & Sons, Ltd.

High-resolution hybrid primitive–conservative upwind schemes for the drift flux model

We consider a drift flux model describing one dimensional isothermal gas–liquid flow in a long pipeline. The model consists of equations for the conservation of mass for each of the phases and the conservation of momentum of the mixture. In addition, an equation – the so-called gas slip relation – is supplied which relates the velocities of the two phases at any point. We present and demonstrate high-resolution hybrid primitive–conservative upwind schemes, which utilize a simple non-conservative MUSCL scheme in shock free flow regions and high-resolution conservative upwind schemes in shocked flow regions. We apply these hybrid schemes to various test problems and compare them with some conservative upwind schemes. The overall conclusion is that the hybrid upwind schemes are as accurate as the corresponding conservative schemes. The hybrid schemes are, however, more efficient.

A Cell-Centered, Finite-Volume Upwind Scheme with Global Conservation

A finite-volume, conservative upwind scheme has been developed, based on the flux-difference splitting method. Pseudo-compressibility is introduced to the continuity equation. The cell-centered is adopted, i. e., nodes for flow variables were placed at the center of each grid cell. With this combination of the scheme and the node-cell layout, the global conservation property has been derived in a straightforward manner.The scheme was applied to two types of flows. First the flow past a circular cylinder was computed using the O-grid at the Reynolds number Re =40. The integrated momentum and mass fluxes at inner and outer boundaries agreed up to more than 9 significant figures after 1, 000 time steps. Thus the global conservation property was confirmed. The computed drag coefficient value agreed well with other computed values. The same flow was computed using the H-grid. The drag coefficient value thus obtained differed very little from the O-grid value.The flow past a flat plate with a point of mapping singularity was computed at an attack angle of 30 degrees. It was confirmed that the global conservation property of the present scheme is not affected by the presence of mapping singularities.

An implicit form for the Osher upwind scheme

Conservative upwind schemes for the Euler equations, such as the Osher scheme, accurately resolve flow discontinuities and correctly model the physics of the problem. However, these schemes require many more arithmetic operations per integration step than simple central-difference schemes and hence result in large computing times. An implicit version of the first-order- and second-order-accurate Osher schemes in two spatial dimensions and generalized coordinates is developed in this study. Because implicit schemes permit the use of large integration steps, in many cases they require fewer integration steps to reach steady-state (especially in calculations on grids with widely varying mesh-cell sizes). The implicit scheme developed in this study accelerated convergence speeds by almost an order of magnitude in the problems considered. Test cases include quasi-one-dimensional nozzle flow and supersonic flow past a cylinder.