Characteristic Galerkin Research Articles

An Extended Finite Element Method for Two-Phase Fluids

An extended finite element method with arbitrary interior discontinuous gradients is applied to two-phase immiscible flow problems. The discontinuity in the derivative of the velocity field is introduced by an enrichment with an extended basis whose gradient is discontinuous across the interface. Therefore, the finite element approximation can capture the discontinuities at the interface without requiring the mesh to conform to the interface, eliminating the need for remeshing. The equations for incompressible flow are solved by a fractional step method where the advection terms are stabilized by a characteristic Galerkin method. The phase interfaces are tracked by level set functions which are discretized by the same finite element mesh and are updated via a stabilized conservation law. The method is demonstrated in several examples.

An efficient characteristic Galerkin scheme for the advection equation in 3-D

Conventional Galerkin-based finite element algorithms perform poorly when modeling advective transport. Here we develop and test a characteristic Galerkin scheme to solve the unsteady three-dimensional advective transport equation. The algorithm uses tracking of the Gaussian quadrature points to project the information from the Eulerian background grid to the Lagrangian grid. Numerical experiments were carried out to investigate the performance of this scheme. Despite speculations to the contrary in the literature, this scheme does not suffer from instability problems. Further, it is conservative, and shows good phase characteristics with slight numerical diffusion. We conclude that the characteristic Galerkin method is a viable and efficient scheme for solving advection problems in three dimensions.

Mathematical analysis of the optimal location of wastewater outfalls

In this work we deal with the design of wastewater treatment systems, mainly the optimal placement of underwater outfalls. This problem can be formulated as a state constrained optimal control problem where the control is the position of the outfalls, the cost function is the sum of the distances to the wastewater farms and the state equations are those modelling dissolved oxygen and biochemical oxygen demand concentrations. We discretize the problem by means of a characteristic Galerkin method and we propose an interior point algorithm for the numerical resolution of the discretized optimization problem.

CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter

AbstractIn this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The second approach is the Galerkin‐least‐squares (GLS) method, in which a least‐squares form of the element residual is added to the basic Galerkin equations. It is shown that both formulations display similar stabilization mechanisms, provided the stabilization parameter of the GLS method is identified with the time step of the CBS approach. This identification can be understood from a formal Fourier analysis of the linearized problem. Copyright © 2001 John Wiley & Sons, Ltd.

Modelling of water pollution in the Thermaikos Gulf with fuzzy parameters

The fuzzy set theory is applied for the definition of the imprecise parameters, which are used in a water pollution model. The physico-chemical coefficients and the loads of pollution sources are expressed in the form of triangular fuzzy numbers. A two-dimensional finite element algorithm, combined with fuzzy logic analysis, is used for the solution of the advection–dispersion equation for ten different water quality variables. The characteristic Galerkin technique was employed for the temporal discretisation, whereas interval operations were conducted for the solution of the algebraic system of equations with fuzzy coefficients. The model was applied for the study of pollution in the Gulf of Thermaikos, located in Northern Greece, for water velocities obtained from a wind-induced circulation model. The concentration distributions of the water quality variables considered are derived in the form of fuzzy numbers, and are presented in relevant diagrams.

A Characteristic Galerkin Method for Discrete Boltzmann Equation

The characteristic Galerkin finite element method for the discrete Boltzmann equation is presented to simulate fluid flows in complex geometries. The inherent geometric flexibility of the finite element method permits the easy use of simple Cartesian variables on unstructured meshes and the mesh clustering near large gradients. The characteristic Galerkin procedure with appropriate boundary condition results in accurate solutions with little numerical diffusion. Several test cases are conducted, including unsteady Couette flows, lid-driven cavity flows, and steady flow past a circular cylinder on unstructured meshes. The numerical results are in good agreement with previous analytical (if applicable), numerical, and experimental results.

A characteristic Galerkin method with adaptive error control for the continuous casting problem

The continuous casting problem is a convection-dominated nonlinearly degenerate diffusion problem. It is discretized implicitly in time via the method of characteristics, and in space via continuous piecewise linear finite elements. A posteriori error estimates are derived for the L 1 L 1 norm of temperature which exhibit a mild explicit dependence on velocity. The analysis is based on special properties of a linear dual problem in non-divergence form with vanishing diffusion and strong advection. Several simulations with realistic physical parameters illustrate the reliability of the estimators and the flexibility of the proposed adaptive method.

Implicit characteristic Galerkin method for convection-diffusion equations

This paper presents a characteristic Galerkin finite element method with an implicit algorithm for solving multidimensional, time-dependent convection–diffusion equations. The method is formulated on the basis of the combination of both the precise and the implicit numerical integration procedures aiming to reference particles. The precise integration procedure with a 2N algorithm is taken as a tool to determine the material (Lagrangian) derivative of the convective function in the operator splitting procedure. The stability analysis of the algorithm and numerical results illustrate good performance of the present method in stability and accuracy. Copyright © 2000 John Wiley & Sons, Ltd.

Implicit characteristic Galerkin method for convection–diffusion equations

Implicit characteristic Galerkin method for convection–diffusion equations

Contaminant transport with non-equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme

A non-equilibrium sorption—advection—diffusion model to simulate miscible pollutant transport in saturated–unsaturated soils is presented. The governing phenomena modelled in the present simulation are: convection, molecular diffusion, mechanical dispersion, sorption, immobile water effect and degradation, including both physical and chemical non-equilibrium processes. A finite element procedure, based on the characteristic Galerkin method with an implicit algorithm is developed to numerically solve the model equations. The implicit algorithm is formulated by means of a combination of both the precise and the traditional numerical integration procedures. The stability analysis of the algorithm shows that the unconditional stability of the present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results illustrate good performance of the present algorithm in stability and accuracy, and in simulating the effects of all the mentioned phenomena governing the contaminant transport and the concentration distribution. Copyright © 2000 John Wiley & Sons, Ltd.

Contaminant transport with non‐equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme

Contaminant transport with non‐equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme

Characteristic Galerkin method for convection-diffusion equations and implicit algorithm using precise integration

This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.

The characteristic-based-split procedure: an efficient and accurate algorithm for fluid problems

In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self-adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi-implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic-based-split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised. Copyright © 1999 John Wiley & Sons, Ltd.

Finite-Element Method for Contaminant Transport in Unsaturated Soils

A numerical model to simulate miscible contaminant transport through unsaturated soils is presented. To account for the influence of multiple nonequilibrium sources on the contaminant transport, six governing phenomena of the miscible contaminant transport (i.e., convection, mechanical dispersion, molecular diffusion, adsorption, degradation, and immobile water effect) are integrated into the present model. The pollutant volumetric concentration in mobile water is taken as primary unknown, whereas the pollutant concentration in immobile water and the solid particles of soils are treated as state variables at the element integration points. Based on a splitting of the generalized convective operator from the diffusive operator, a modified version of the characteristic Galerkin method is developed to discretize the equations governing the contaminant transport phenomena. A fully explicit algorithm is then derived for the numerical solution of the finite-element equations in time domain. The numerical examples illustrate the performance and the capability of the presented model and algorithms.

Numerical Solution of the Incompressible Navier–Stokes Equations with Coriolis Forces Based on the Discretization of the Total Time Derivative

In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on a finite difference discretization in time and a finite element discretization in space. When the viscosity is very small, numerical oscillations may appear due both to the high Reynolds number and to the presence of the Coriolis forces. To overcome these oscillations, a special discretization in time is proposed. The idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system. After this is done, a further high-order approximation is introduced, leading to a problem posed in the rotating frame of reference and in spatial coordinates. In contrast with the straightforward discretization of the original equations, some additional terms appear that enhance the stability of the numerical scheme. In the absence of Coriolis forces, the method is a generalization of the characteristic Galerkin technique for convection-dominated flows.

Multidimensional characteristic Galerkin methods for hyperbolic systems

We consider characteristic Galerkin methods for the solution of hyperbolic systems of partial differential equations of first order. A new recipe for the construction of approximate evolution operators is given in order to derive consistent methods. With the help of semigroup theory we derive error estimates for classes of characteristic Galerkin methods. The theory is applied to the wave equation and also to the Euler equations of gas dynamics. In the latter case one can show that Fey's genuinely multidimensional method can be reinterpreted as a characteristic Galerkin method. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Characteristic Galerkin Schemes for Scalar Conservation Laws in Two and Three Space Dimensions

In this paper we are concerned with a finite element method for multidimensional scalar conservation laws: we describe a general formulation of the Euler characteristic Galerkin (ECG) scheme, motivated by key features of the one-dimensional ECG scheme. The method is defined by projecting the transport collapse operator onto a finite element space spanned by piecewise constant basis functions. The ECG scheme is {em TVD, monotone, maximum norm nonincreasing and unconditionally stable in the $L^{1}$-norm}. To the best of our knowledge, there is no other method which has all these properties. However, with piecewise constant basis functions, the ECG scheme is at most first-order accurate; greater accuracy can be obtained through a recovery procedure. Unlike conventional dimension-by-dimension methods, multidimensional ECG schemes contain important terms which describecorner effects. Nonuniform meshes and large time steps can be used with this class of schemes.

Characteristic Petrov-Galerkin Schemes for 2-D Conservation Laws

Characteristic Galerkin (CG) schemes were first presented for one- and two-dimensional conservation laws by Morton. In this paper, a class of modified second order accurate characteristic Petrov-Galerkin (CPG) schemes obtained by combining a unit leap function with a linear element as the try function, called a nonlinear element, are derived for one- and two-dimensional conservation laws. The schemes constructed are due to characteristic relations and the structure of the solutions, and made to have high resolution and nonoscillations. Finally, numerical experiments are given.

Euler characteristic Galerkin scheme with recovery

Cet article presente la formulation generale du schema de Galerkin utilisant les caracteristiques (ECG) pour les lois de conservation scalaires et dans le cadre de la theorie des integrales de Riemann et Stieltjes. Nous montrons que ce schema est equivalent a l'operateur de transport-collapse de Brenier. Pour obtenir une plus grande precision, nous envisageons deux types de reconstruction, suivant que sont utilisees des fonctions lineaires continues ou des fonctions lineaires discontinues. La convergence du schema numerique est alors montree, obtenue apres avoir obtenu certaines majorations. Nous prouvons finalement que la limite de la fonction approchee, obtenue apres reconstruction, est une solution admissible des equations de conservation

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.