Taylor-Galerkin Methods Research Articles

1D Numerical modelling of dam break using finite element method

In numerical modeling, dam break is one case that has its own challenges, because shock wave is found in the dam break modeling that usually provides a numerical instability. Usually, dam break problem is solved by Saint Venant equation using a finite difference method with artificial dissipation or Total Variation Diminishing (TVD) filter. But in this research, finite element method and the finite difference method are used. To verify the accuracy of the model, a comparison against the Stoker analytical method for dam break case was performed. Numerical modeling of dam break is required to find out the collapse area, thus it is used for determining mitigation that can be done in the area, related to dam safety. In numerical modeling, oscillation or numerical instability often occurs, for which special treatment is required to reduce or eliminate the oscillations. In this research, the treatment for that case is a Hansen filter for both methods. From the simulation result, it is found that Hansen filter is sensitive in reducing oscillation depending on the correction factor value and Δt that used. For dam break case, after filter applied, the value of Pearson Correlation Coefficient of Taylor Galerkin and Mac-Cormack methods are 0.999. The error rate for a Taylor Galerkin method are 0.118% at t = 3s and 0.123% at t = 10s. The error rate for Mac-Cormack method are 0.043% at t = 3s and 5.048% at t = 10s. From the comparison of the model, it can be concluded that Taylor Galerkin finite element method proved to be capable and more accurate in simulating dam break compared to Mac-Cormack finite difference method.

Thermal–fluid transient two-dimensional characteristic-based-split finite-element model of a distribution transformer

This study presents a two-dimensional (2D) finite-element (FE) linear model for predicting the thermal and fluid dynamics of an oil-filled distribution transformer. The model is based on the solution of the Navier–Stokes and Maxwell equations which are weakly coupled. Two electromagnetic FE models are used to calculate the load and no load losses of the transformer. The thermal–fluid equations are solved using first-order 2D FE and the characteristic-based-split (CBS) scheme. The CBS is used for the first time in the thermal fluid modelling of a transformer thereby is the proposed contribution of this study. The CBS algorithm is applicable to all ranges of flow and gives the same results as specialised methods such as the Petrov–Galerkin and Taylor–Galerkin methods. A natural convection analysis for an incompressible Newtonian flow in a distribution transformer is presented. Convection, radiation and symmetry boundary conditions are taken into account in the model. The model predictions are compared against those obtained with a commercial software and good agreement was obtained. The proposed model was programmed in C language.

Wall distance calculation using the Eikonal/Hamilton‐Jacobi equations on unstructured meshes

PurposeThe purpose of this paper is to numerically solve Eikonal and Hamilton‐Jacobi equations using the finite element method; to use both explicit Taylor Galerkin (TG) and implicit methods to obtain shortest wall distances; to demonstrate the implemented methods on some realistic problems; and to use iterative generalized minimal residual method (GMRES) method in the solution of the equations.Design/methodology/approachThe finite element method along with both the explicit and implicit time discretisations is employed. Two different forms of governing equations are also employed in the solution. The Eikonal equation in its original form is used in the explicit Taylor Galerkin discretisation to save computational time. For implicit method, however, the convection‐diffusion form in its conservation form is used to maintain spatial stability.FindingsThe finite element solution obtained is both accurate and smooth. As expected the implicit method is much faster than the explicit method. Though the proposed finite element solution procedures in serial is slower than the standard search procedure, they are suitable to be used in a parallel environment.Originality/valueThe finite element procedure for Eikonal and Hamilton‐Jacobi equations are attempted for the first time. Though the finite volume and finite difference‐based computational fluid dynamics (CFD) solvers have started employing differential equations for wall distance calculations, it is not common for finite element solvers to use such wall distance calculations. The results presented here clearly show that the proposed methods are suitable for unstructured meshes and finite element solvers.

One-step Taylor–Galerkin methods for convection–diffusion problems

Third and fourth order Taylor–Galerkin schemes have shown to be efficient finite element schemes for the numerical simulation of time-dependent convective transport problems. By contrast, the application of higher-order Taylor–Galerkin schemes to mixed problems describing transient transport by both convection and diffusion appears to be much more difficult. In this paper we develop two new Taylor–Galerkin schemes maintaining the accuracy properties and improving the stability restrictions in convection–diffusion. We also present an efficient algorithm for solving the resulting system of the finite element method. Finally we present two numerical simulations that confirm the properties of the methods.

TIME ACCURATE FAST THREE-STEP WAVELET-GALERKIN METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

We introduce the concept of three-step wavelet-Galerkin method based on the Taylor series expansion in time. Unlike the Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving nonlinear problems. Numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of functions which are smooth, except for localized regions, up to any given accuracy are presented. Here numerical experiments deal with advection equation with the spiky solution in one dimension, two dimensions and nonlinear equation with a shock in solution in two dimensions. Numerical results indicate the versatility and effectiveness of the proposed scheme.

A three-step wavelet Galerkin method for parabolic and hyperbolic partial differential equations

A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and O(2−jp ) accurate in space. Unlike Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving non-linear problems. The compactly supported orthogonal wavelet bases D6 developed by Daubechies are used in the Galerkin scheme. The proposed scheme is tested with both parabolic and hyperbolic partial differential equations. The numerical results indicate the versatility and effectiveness of the proposed scheme.

Wavelet multilayer Taylor Galerkin schemes for hyperbolic and parabolic problems

In this study wavelet based high-order Taylor Galerkin methods is introduced. Additional time layers are used to obtain high-order temporal accuracy unattainable within a two-step strategy. Two explicit model schemes are constructed and asymptotic stability of schemes are verified. The compactly supported orthogonal wavelet bases developed by Daubechies are used in Galerkin scheme. The behavior of the proposed schemes is illustrated through one and two dimensional problems.

Time-Domain Methods for Solving Problems of Linear Acoustics

This paper describes a class of high-order Taylor-Galerkin methods for solving problems of linear acoustics characterized by the classical (scalar) wave equation. The methods provide high-order temporal accuracy and unconditional stability on arbitrary nonuniform finite element meshes of varying element sizes and shape functions.

On solving structural acoustics problems in time-domain

The structural acoustics problem is formulated as a hyperbolic system of conservation laws which leads to an abstract Cauchy problem in common Hilbert space settings. The Cauchy problem is approximated by using high-order, multi-stage Taylor-Galerkin methods which provide high-order temporal accuracy and unconditional stability on arbitrary (unstructured) finite element grids. The formulation is extended to problems posed on unbounded domains by introduction of iterative radiation boundary conditions. The proposed approaches are shown to produce very good results for the test cases considered.

Multilayer Taylor-Galerkin schemes for convection problems

A new approach to derivation of high-order Taylor–Galerkin methods is introduced. Additional time layers are shown to yield temporal accuracy unattainable within a two-layer strategy. Two explicit model schemes are constructed, and the von Neumann stability analysis is conducted. The behaviour of the proposed schemes is illustrated on one- and two-dimensional benchmark problems. Copyright © 1999 John Wiley & Sons, Ltd.

Stabilised finite element methods for steady incompressible flow

We give a brief overview of frequently used stabilised finite element schemes for steady incompressible flow. We derive Taylor-Galerkin schemes for the flow equations and show how SUPG-type and Taylor-Galerkin methods can be interpreted within the same framework.

A simple scheme for the finite element solution of a pure advection equation

The present paper deals with the finite element solution of an advection equation. The method of characteristic lines is combined with the finite element method. Three interpolating functions are employed: a cubic, a quadratic and a linear polynomials. The proposed scheme has two key features. One is the simplicity of its algorithm. The other is the combination of the proposed algorithm with the limiting procedure for the purpose of the suppression of numerical oscillations. Three interpolating functions are tested through four numerical examples: the advection of a box-shaped profile, the advection of an elliptic profile, the rotation of a cosine-shaped hill, and the advection of a square-shaped hill. The effectiveness of the proposed scheme is shown by comparing the present numerical results with other popular finite element schemes, i.e., SUPG, Taylor-Galerkin methods and so on. In the present test calculations, the scheme with a quadratic interpolating function has given the best results. The computing time of the proposed scheme is much faster than that of other well-known finite element schemes.

Tensor Universal Serendipity Elements and unsteady Taylor-Galerkin finite element method

A new kind of Universal Serendipity Element (USE)— the Tensor Universal Serendipity Element (TUSE) is constructed by using both tensor force finite elements and the basic idea of USE. The formulation of shape functions and their derivatives for TUSE is presented. TUSE can be used to study steady and unsteady transonic flow fields when combined with Taylor-Galerkin Finite Element Methods, the NND scheme in FDM, and four-stage Runge-Kutta methods. As numerical examples the transonic flow in cascades and one kind of complex unsteady transonic axisymmetric flow in engineering are studied. It is shown that the algorithm presented in this paper is efficient and robust.

Analysis of finite element schemes for convection‐type problems

AbstractVarious finite element schemes of the Bubnov–Galerkin and Taylor–Galerkin types are analysed to obtain the expressions of truncation errors. This way, dispersion errors in the transient, and diffusion errors both in the transient and in the steady state, are identified. Then, with reference to the transient advection–diffusion equation, stability limits are determined by means of a general von Neumann procedure. Finally, the operational equivalence between Taylor–Galerkin methods, utilized for pseudo‐transient calculations, and Petrov–Galerkin methods, derived for the steady state forms of the advection–diffusion equation, is illustrated. Theoretical conclusions are supported by the results of numerical experiments.

Méthodes d'éléments finis pour les problèmes de convection-diffusion

ABSTRACT This paper presents a brief overview of generalized Galerkin methods for the finite element solution of transport problems governed by convection and diffusion. The first part is concerned with steady problems. The emphasis is placed upon the use of Petrov-Galerkin methods to reproduce in the finite element context the upwind effect previously used with success in finite differences. The second part deals with transient problems describing pure convection and, more generally, with situations governed by first-order hyperbolic equations. The methods discussed include techniques making an explicit use of the characteristic curves, Petrov-Galerkin and Taylor-Galerkin methods, as well as least-squares methods.

Generalized Galerkin Methods for Convection Dominated Transport Phenomena

A brief survey is made of recent advances in the development of finite element methods for convection dominated transport phenomena. Because of the nonsymmetric character of convection operators, the standard Galerkin formulation of the method of weighted residuals does not possess optimal approximation properties in application to problems in this class. As a result, numerical solutions are often corrupted by spurious node-to-node oscillations. For steady problems describing convection and diffusion, spurious oscillations can be precluded by the use of upwind-type finite element approximations that are constructed through a proper Petrov-Galerkin weighted residual formulation. Various upwind finite element formulations are reviewed in this paper, with a special emphasis on the major breakthroughs represented by the so-called streamline upwind Petrov-Galerkin and Galerkin least-squares methods. The second part of the paper is devoted to a review of time-accurate finite element methods recently developed for the solution of unsteady problems governed by first-order hyperbolic equations. This includes Petrov-Galerkin, Taylor-Galerkin, least-squares, and various characteristic Galerkin methods. The extension of these methods to deal with unsteady convection-diffusion problems is also considered.

An analysis of time discretization in the finite element solution of hyperbolic problems

The problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined. A modified equation analysis reveals that the classical, second-order accurate, time-stepping algorithms, i.e., the Lax-Wendroff, leap-frog, and Crank-Nicolson methods, properly combine with piecewise linear finite elements in advection problems only for small values of the time step. On the contrary, as the Courant number increases, the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit CFL property are not achieved. These fundamental numerical properties can, however, be recovered, while still remaining in the standard Galerkin finite element setting, by increasing the order of accuracy of the time discretization. This is accomplished by exploiting the Taylor series expansion in the time increment up to the third order before performing the Galerkin spatial discretization using piecewise linear interpolations. As a result, it appears that the proper finite element equivalents of second-order finite difference schemes are implicit methods of incremental type having third- and fourth-order global accuracy on uniform meshes (Taylor-Galerkin methods). Numerical results for several linear examples are presented to illustrate the properties of the Taylor-Galerkin schemes in one- and two-dimensional calculations.

Generalised galerkin methods for hyperbolic problems

We consider time-accurate methods for unsteady problems on a fixed mesh. Important advantages of the Galerkin formulation are lost as the time step is increased, and we explore first attempts with Petrov-Galerkin and Taylor-Galerkin methods to improve matters. This leads naturally to the study of an important class of methods which we call characteristic Galerkin methods. Some of the many possibilities offered by these are described and a particular algorithm is developed and illustrated on a number of test problems in one dimension.

Time-accurate solution of advection-diffusion problems by finite elements

The Taylor-Galerkin method, recently proposed for the spatial and temporal discretization of hyperbolic equations, is employed to derive accurate and efficient numerical schemes for the solution of time-dependent advection-diffusion problems. Two distinct numerical strategies are discussed: the first is suitable for evolutionary problems, while the second is appropriate for situations in which a steady state is eventually reached. For purely evolutionary problems the Taylor-Galerkin method is applied to the complete advection-diffusion equation and, through modifications of the standard ‘mass’ matrix, is shown to generate an incremental form of the Crank-Nicolson time-stepping method. Such an incremental form also lends itself to the derivation of computationally simple, but accurate, explicit algorithms. To deal with transient situations which evolve towards a highly convective steady state, the ‘global’ Taylor-Galerkin method is found to be ineffective since it reduces to the standard Galerkin formulation as the temporal term vanishes. For such cases we suggest the use of a splitting-up method in which advection and diffusion are treated separately by appropriate Taylor-Galerkin methods. The proposed methods are illustrated first for linear, constant-coefficient equations. They are successively extended to deal with nonlinear and multi-dimensional problems. Numerical applications indicate their effectiveness.

Recent advances in computational methods for steady and transient transport problems

A survey has been made of some recent developments in computational methods for steady and transient transport problems describing convection and diffusion. The emphasis is placed on the use of finite element methods since this is an area where many recent developments have taken place. Various generalizations of the standard Galerkin formulation are considered which enable it to deal successfully with convection-dominated transport processes. These include both Petrov-Galerkin and Taylor-Galerkin methods.