Quang-Huy Tran

A relaxation method for two-phase flow models with hydrodynamic closure law

Summary.This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided.

A Domain Decomposition Method for the Acoustic Wave Equation with Discontinuous Coefficients and Grid Change

A domain decomposition technique is proposed for the computation of the acoustic wave equation in which the bulk modulus and density fields are allowed to be discontinuous at the interfaces. Inside each subdomain, the method presented coincides with the second-order finite difference schemes traditionally used in geophysical modeling. However, the possibility of assigning to each subdomain its own space-step makes numerical simulations much less expensive. Another interest of the method lies in the fact that its hybrid variational formulation naturally leads to exact equations for gridpoints on the interfaces. Transposing Babuska--Brezzis formalism on mixed and hybrid finite elements provides a suitable functional framework for this domain decomposition formulation and shows that the inf-sup condition remains the basic requirement for convergence to occur.

A Semi-implicit Relaxation Scheme for Modeling Two-Phase Flow in a Pipeline

This work is devoted to the numerical approximation of the solutions to the system of conservation laws which arises in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic law. We have previously proposed an explicit relaxation scheme which allows us to cope with these nonlinearities. But the system considered has eigenvalues which are of very different orders of magnitude, which prevents the explicit scheme from being effective, since the time step has to be very small. In order to solve this effectiveness problem, we now proceed to construct a scheme which is explicit with respect to the small eigenvalues and linearly implicit with respect to the large eigenvalues. Numerical evidences are provided.

Local Time Stepping Applied to Implicit-Explicit Methods for Hyperbolic Systems

In the context of systems of nonlinear conservation laws it can be crucial to use adaptive grids in order to correctly simulate the singularities of the solution over long time ranges while keeping the computing time within acceptable bounds. The adaptive space grid must vary in time according to the local smoothness of the solution. More sophisticated and recent methods also adapt the time-step locally to the space discretization according to the stability condition. We present here such a method designed for an explicit-implicit Lagrange-projection scheme, addressing physical problems where slow kinematic waves coexist with fast acoustic ones. Numerical simulations are presented to validate the algorithms in terms of robustness and efficiency.

Multiresolution technique and explicit-implicit scheme for multicomponent flows

In the context of multicomponent flows, we are faced with PDE systems solutions com- bining waves whose speeds are several orders of magnitude apart. Of these waves, only the slow kinematic ones that represent transport phenomena are of concern to us. The fast acoustic ones, al- though uninteresting for the physical application conside red, nevertheless impose a prohibitively small time-step (via the classical CFL restriction) if treated ex plicitly. This is why we propose to use a hy- brid finite-volume scheme in which fast waves are handled by a linearized implicit formulation and slow waves remain explicitly solved. To further decrease the computational cost, mostly due to the complexity of nonlinear thermodynamical laws, we combine this method with a fully adaptive mul- tiresolution scheme. At each time step, a multiscale analysis followed by the thresholding of small details enables us to discretize the solution over a time-va rying adaptive grid, based on the smooth- ness of the relevant phenomenon. Particular attention is gi ven to the extension of the reference scheme to non uniform grid and to the prediction strategy of the adap tive grid from one time-step to another. Finally, efficiency in terms of computing time requirements is studied in conjunction with the accuracy performances.

A RELAXATION METHOD VIA THE BORN–INFELD SYSTEM

The semilinear relaxation was introduced by Jin and Xin [Comm. Pure Appl. Math.48, 235, (1995)] in order to approximate the conservation law ∂tu + ∂xf(u) = 0 for any flux function f ∈ 𝒞1 (ℝ;ℝ). In this paper, we propose an alternative relaxation technique for scalar conservation laws of the form ∂tu + ∂xu(1 - u)g(u) = 0, where g ∈ 𝒞1 ([0, 1]; ℝ) and 0 ∉ g(]0, 1[). We extend this new philosophy to an arbitrary flux function f whenever possible. Unlike the semilinear approach, the new relaxation strategy does not involve any tuning parameter, but makes use of the Born–Infeld system. Another advantage of this method is that it enables us to achieve a maximum principle on the velocities w = (1 - u)g and z = -ug, which turns out to be a physically interesting and helpful feature in the context of some two-phase flow problems.

Local Time Stepping for Implicit-Explicit Methods on Time Varying Grids

In the context of nonlinear conservation laws a model for multiphase flows where slow kinematic waves co-exist with fast acoustic waves is discretized with an implicit-explicit time scheme. Space adaptivity of the grid is implemented using multiresolution techniques and local time stepping further enhances the computing time performances. A parametric study is presented to illustrate the robustness of the method.

Large Time Step Positivity-Preserving Method for Multiphase Flows

Using a relaxation strategy in a Lagrangian-Eulerian formulation, we propose a scheme in local conservation form for approximating weak solutions of complex compressible flows involving wave speeds of different orders of magnitude. Explicit time integration is performed on slow transport waves for the sake of accuracy while fast acoustic waves are dealt with implicitly to enable large time stepping. A CFL condition based on the slow waves is derived ensuring positivity properties on the density and the mass fraction. Numerical benchmarks validate the method.

Semi-Implicit Multiresolution for Multiphase Flows

In the context of multiphase flows we are faced with vector PDE solutions combining waves whose speeds are several orders of magnitude apart. The wave of interest is the transport one, and is relatively slow. The other fast acoustic waves are not interesting but impose a very restrictive CFL condition if a fully explicit in time scheme is considered. We therefore use a time semi-implicit conservative scheme where the fast waves are handled with a linearized implicit formulation and the slow wave remains explicitly solved. The CFL condition, governed by the explicit wave speed is then optimal. We combine this method with a multiscale analysis of the vector solution which enables to use a time varying adaptive grid based on the relevant smoothness properties of the discrete solution. In this short paper we compare different strategies to evaluate the fluxes at cells interfaces on a non uniform grid.

Propagation and attenuation of a Ricker pulse in one-dimensional randomly heterogeneous media: a numerical study

Abstract Numerical simulations are extensively used to study some major features of acoustic wave propagation in one-dimensional random media. Gaussian and exponential stationary media, in which a Ricker pulse is applied, are considered. The properties numerically shown in this paper deal with the irregular progression of the wave front and the changes in its energy distribution with respect to time. Particular emphasis is given to the study of the apparent attenuation factor. Not only are the results shown to be in good agreement with existing analytical studies, but they also provide a more thorough insight into propagation phenomena, especially when the corresponding analytical predictions no longer hold true.