Predictor Multi-corrector Algorithm Research Articles

A fractional-step method for the incompressible Navier-Stokes equations related to a predictor-multicorrector algorithm

SUMMARY An implicit fractional-step method for the numerical solution of the time-dependent incompressible Navier–Stokes equations in primitive variables is studied in this paper. The method, which is first-orderaccurate in the time step, is shown to converge to an exact solution of the equations. By adequately splitting the viscous term, it allows the enforcement of full Dirichlet boundary conditions on the velocity in all substeps of the scheme, unlike standard projection methods. The consideration of this method was actually motivated by the study of a well-known predictor–multicorrector algorithm, when this is applied to the incompressible Navier–Stokes equations. A new derivation of the algorithm in a general setting is provided, showing in what sense it can also be understood as a fractional-step method; this justifies, in particular, why the original boundary conditions of the problem can be enforced in this algorithm. Two different finite element interpolations are considered for the space discretization, and numerical results obtained with them for standard benchmark cases are presented. © 1998 John Wiley & Sons, Ltd.

A symmetric formulation for computing transient shallow water flows

The shallow water equations fall into the category of symmetric advective-diffusive systems with source terms. These types of equations can be very effectively solved using stabilized methods, such as SUPG and GLS. A semi-discrete finite element method based on these ingredients is presented for the computation of transient shallow water flows. Special care has been taken in the design of the operators in order to improve the performance for unsteady calculations. The solution is advanced in time via a predictor multi-corrector algorithm which includes as special cases the second-order trapezoidal rule, a first-order ‘explicit’ method and a first-order implicit method, equivalent to the constant-in-time element of the space-time formulation.

Parallel Finite Element Simulation of Miscible Displacements in Porous Media

Abstract In this work, parallel finite element techniques for the simulation of miscible displacements in porous media are addressed. The pressure equation is approximated by Galerkin's method and the velocity field computed through a post-processing approach to recover the required accuracy. The concentration equation is approximated in space by the Streamline Upwind Petrov Galerkin (SUPG) plus a discontinuity-capturing operator. The resulting semi-discrete equations are approximated in time by a predictor-multicorrector algorithm. To increase the robustness of our solution scheme, the feedback control theory is used to select the proper time-step. The pressure, velocity and concentration linear systems of equations are solved with parallel element-by-element iterative techniques. Performance measurements on two different machines, the CRAY Y-MP and the CRAY C90, for a five-spot pattern with high mobility ratio on random heterogeneous media are presented to show that the numerical techniques employed are accurate and result in a fast code.

A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations

A space-time element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms and boundary conditions. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. Numerical results are presented to demonstrate the performance of the method.

A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms

A Fourier stability and accuracy analysis of the space-time Galerkin/least squares method as applied to a time-dependent advective-diffusive model problem is presented. Two time discretizations are studied: a constant-in-time approximation and a linear-in-time approximation. Corresponding space-time predictor multi-corrector algorithms are also derived and studied. The behavior of the space-time algorithms is compared to algorithms based on semidiscrete formulations.

Numerical Simulation of Deep‐Well Wet Oxidation Reactor using Steam

A finite element model is developed for the numerical simulation of the start-up of a deep-well wet oxidation reactor using steam. The model is capable of handling different cases of start-up, using saturated steam of arbitrary quality including hot water. The transient temperature field in the well-earth system is studied. The governing equation for the earth is the conductive heat equation. For the reactor, the analysis involves solving the energy balance equations describing the convective and intertube heat transfer, mass balance equations, and thermodynamic relations for the fluid in the tubes. The equations for the reactor and the earth are coupled by the continuity of the temperature and heat flux at the interface between them. A Galerkin finite element formulation is used for the spatial discretization of the heat equation in the earth; a Petrov-Galerkin finite element formulation is employed for the energy balance equations in the reactor tubes. The resulting set of ordinary differential equations is discretized in time and solved by a predictor-multicorrector algorithm. The model is tested on a typical deep-well reactor to study its start-up dynamics. The results can be used to estimate the start-up time and the amount of steam or water required to obtain the necessary initiation temperature for the process.

Nonlinear Transient Phenomena in Elastic-Plastic Solids

A finite element model is proposed to analyze nonlinear transient phenomena in elastic-plastic solids due to both material and geometrical nonlinearities. Time integration of the semi discrete finite element equations is performed by using an implicit-explicit predictor-multicorrector algorithm developed by Hughes and co-workers. It is shown that the algorithm can be simply and concisely implemented. The technique allows for a convenient selection of implicit and explicit elements, and for a convenient implicit-explicit split of the various operators appearing in the equations of motion. The procedure proves to be extremely effective in dealing with incompressible plastic flow problems. Numerical results which demonstrate the versatility and accuracy of the method are presented.