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Spurious Oscillations (spurious + oscillation)
Selected AbstractsWeighted average flux method and flux limiters for the numerical simulation of shock waves in rigid porous mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2002R. Torrens Abstract The one-dimensional flow field generated by the passage of a shock wave in a rigid, thermoelastic porous foam has been simulated using a two-phase mathematical model. The work presented here makes use of the weighted average flux method to solve the system of six equations that govern the problem. Spurious oscillations are eliminated through the application of total variation diminishing limiting methods. Four different limiters were tested: van Leer, SuperA, MinA and van Albada. Numerical tests were carried out to verify the performance of each flux limiter in terms of accuracy. The results were compared to analytical and previously obtained data to assess the performance of the mathematical model. Excellent agreement was obtained. Copyright © 2002 John Wiley & Sons, Ltd. [source] Higher-resolution convection schemes for flow in porous media on highly distorted unstructured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2008Sadok Lamine Abstract Higher-resolution schemes are presented for convective flow approximation on highly distorted unstructured grids. The schemes are coupled with continuous full-tensor Darcy-flux approximations. A sequence of non-uniform and distorted grid formulations are developed and compared for a range of unstructured meshes with variable grid spacing. The higher-order schemes are constructed using non-uniform grid slope limiters such that they are stable with a local maximum principle, ensuring that solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full-tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher-order formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts. Copyright © 2008 John Wiley & Sons, Ltd. [source] An assumed-gradient finite element method for the level set equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2005Hashem M. Mourad Abstract The level set equation is a non-linear advection equation, and standard finite-element and finite-difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed-distance function. For some interface-evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity-capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed-gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level-set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton,Jacobi equation with convex/non-convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite-element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd. [source] Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation lawsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2009Ritesh Kumar Dubey Abstract In this work we present an upwind-based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non-linear problems by using the framework of relaxation system that converts a non-linear conservation law into a system of linear convection equations with a non-linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non-standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd. [source] A higher-order predictor,corrector scheme for two-dimensional advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Chuanjian Man Abstract A higher-order accurate numerical scheme is developed to solve the two-dimensional advection,diffusion equation in a staggered-grid system. The first-order spatial derivatives are approximated by the fourth-order accurate finite-difference scheme, thus all truncation errors are kept to a smaller order of magnitude than those of the diffusion terms. Therefore, there is no need to add an artificial diffusion term to balance the unwanted numerical diffusion. For the time derivative, the fourth-order accurate Adams,Bashforth predictor,corrector method is applied. The stability analysis of the proposed scheme is carried out using the Von Neumann method. It is shown that the proposed algorithm has good stability. This method also shows much less spurious oscillations than current lower-order accurate numerical schemes. As a result, the proposed numerical scheme can provide more accurate results for long-time simulations. The proposed numerical scheme is validated against available analytical and numerical solutions for one- and two-dimensional transport problems. One- and two-dimensional numerical examples are presented in this paper to demonstrate the accuracy and conservative properties of the proposed algorithm by comparing with other numerical schemes. The proposed method is demonstrated to be a useful and accurate modelling tool for a wide range of transport problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] PRICE: primitive centred schemes for hyperbolic systemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2003E. F. Toro Abstract We present first- and higher-order non-oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non-conservative) form. Non-conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non-conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second-order non-oscillatory methods. We have used the MUSCL-Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher-order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non-conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind-based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd. 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