Oscillation limits for weighted residual methods applied to convective diffusion equations

Abstract

Convection-diff usion equations are difficult to solve when the convection term dominates because most solution methods give solutions which oscillate in space. Previous criteria b a e d on the one-dimensional convection-diffusion equation have shown that finite difference and Galerkin (linear or quadratic basis functions) will not give oscillatory solutions provided the Peclet number times the mesh size (Pe A x ) is below a critical value. These criteria are based on the solution at the nodes, and ensure that the nodal values are monotone. Similar criteria are developed here for other methods: quadratic Galerkin with upwind weighting, cubic Galerkin, orthogonal collocation on finite elements with quadratic, cubic or quartic polynomials using Lagrangian interpolation, cubic or quartic polynomials using Hermite interpolation, and the method of moments. The nodal values do not oscillate for collocation or moments methods with Hermite cubic polynomials regardless of the value of Pe Ax.A new criterion is developed for all methods based on the monotonicity of the solutions throughout the domain. This criterion is more restrictive than one based only on the nodal values. All methods that are second order (Ax') or better in truncation error give oscillatory solutions (based on the entire domain) unless Pe Ax is below a critical value. This value ranges from 2 for finite difference methods to 4.6 for Hermite, quartic, collocation methods.PeAx =z 2 Christie er a1.' provided a criterion for the one-dimensional steady-state equation with the Galerkin finite element method and linear or quadratic trial functions. Linear functions give the criterion (l), whereas quadratic functions give the criterion Pehx s 4(2)

Environmental Hazards Associated with a Fuel Leakage in a Confined Space of Air

“…The physical details of these processes are not adequately known and most modelling attempts, though involving complicated numerical methods, could claim only approximate solutions of the problem [16]. Quantitative estimates of the flow fields and patterns and multidimensional concentration profiles are beyond the scope of this paper.…”

Modeling of Subsurface Biobarrier Formation

“…When Pe>>1, sharp fronts and plumes remain sharp and cause numerical difficulties. Typically, the criterion for oscillation-free solution requires that the grid Peclet number Pe = Pe∆x/L = O(1) (Jensen and Finlayson, 1980). However, in underground flows with field-scale pressure gradients applied by pumping wells, Peclet numbers greater than 10 2 are common (Lake and Hirasaki, 1981), so the near-hyperbolic regime is important in engineering applications.…”

Comparación de distintos esquemas numéricos para resolver la onda cinemática

“…Dentro de las posibles formulaciones se han desarrollado los esquemas correspondientes al procedimiento estándar de Galerkin (Vieux, 1988), considerando interpolación lagrangiana lineal (Berger y Stockstill, 1995), cuadrática (Muñoz-Carpena, Miller y Parson, 1993) y cúbica; e interpolación mediante polinomios de Hermíte de 3 er grado (Jensen y Finlayson, 1980). Así mismo se ha analizado el comportamiento de las diferentes soluciones obtenidas.…”