497 research outputs found
Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media
In this paper, we study the unconditional convergence and error estimates of
a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete
scheme for the equations of incompressible miscible flow in porous media. We
prove that the optimal error estimates hold without any time-step
(convergence) condition, while all previous works require certain time-step
condition. Our theoretical results provide a new understanding on commonly-used
linearized schemes for nonlinear parabolic equations. The proof is based on a
splitting of the error function into two parts: the error from the time
discretization of the PDEs and the error from the finite element discretization
of corresponding time-discrete PDEs. The approach used in this paper is
applicable for more general nonlinear parabolic systems and many other
linearized (semi)-implicit time discretizations
Unified convergence analysis of numerical schemes for a miscible displacement problem
This article performs a unified convergence analysis of a variety of
numerical methods for a model of the miscible displacement of one
incompressible fluid by another through a porous medium. The unified analysis
is enabled through the framework of the gradient discretisation method for
diffusion operators on generic grids. We use it to establish a novel
convergence result in of the approximate
concentration using minimal regularity assumptions on the solution to the
continuous problem. The convection term in the concentration equation is
discretised using a centred scheme. We present a variety of numerical tests
from the literature, as well as a novel analytical test case. The performance
of two schemes are compared on these tests; both are poor in the case of
variable viscosity, small diffusion and medium to small time steps. We show
that upstreaming is not a good option to recover stable and accurate solutions,
and we propose a correction to recover stable and accurate schemes for all time
steps and all ranges of diffusion
An HMM--ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media
We design a numerical approximation of a system of partial differential
equations modelling the miscible displacement of a fluid by another in a porous
medium. The advective part of the system is discretised using a characteristic
method, and the diffusive parts by a finite volume method. The scheme is
applicable on generic (possibly non-conforming) meshes as encountered in
applications. The main features of our work are the reconstruction of a Darcy
velocity, from the discrete pressure fluxes, that enjoys a local consistency
property, an analysis of implementation issues faced when tracking, via the
characteristic method, distorted cells, and a new treatment of cells near the
injection well that accounts better for the conservativity of the injected
fluid
Discontinuous Galerkin methods for convection-diffusion equations and applications in petroleum engineering
This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in Linfinity(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in [J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics (2019)] showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out
HIGH ORDER BOUND-PRESERVING DISCONTINUOUS GALERKIN METHODS AND THEIR APPLICATIONS IN PETROLEUM ENGINEERING
This report contains researches in the theory of high-order bound-preserving (BP) discontinuous Galerkin (DG) method and their applications in petroleum engineering. It contains both theoretical analysis and numerical experiments. The compressible miscible displacements and wormhole propagation problem, arising in petroleum engineering, is used to describe the evolution of the pressure and concentrations of different components of fluid in porous media. The important physical features of concentration and porosity include their boundedness between 0 and 1, as well as the monotone increasing for porosity in wormhole propagation model. How to keep these properties in the simulation is crucial to the robustness of the numerical algorithm. In the first project, we develop high-order bound-preserving discontinuous Galerkin methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the jth component,cj, should be between 0 and 1. The main idea is stated as follows. First, we apply the second-order positivity-preserving techniques to all concentrations c′ js and enforce P jcj= 1 simultaneously to obtain physically relevant boundedness for every components. Then, based on the second-order BP schemes, we use the second-order numerical fluxes as the lower order one to combine with high-order numerical fluxes to achieve the high-order accuracy. Finally, since the classical slope limiter cannot be applied to polynomial upper bounds, we introduce a new limiter to our algorithm. Numerical experiments are given to demonstrate the high-order accuracy and good performance of the numerical technique. In our second project, we propose high-order bound-preserving discontinuous Galerkin methods to keep the boundedness for the porosity and concentration of acid, as well as the monotone increasing for porosity. The main technique is to introduce a new variable r to replace the original acid concentration and use a consistent flux pair to deduce a ghost equation such that the positive-preserving technique can be applied on both original and deduced equations. A high-order slope limiter is used to keep a polynomial upper bound which changes over time for r. Moreover, the high-order accuracy is attained by the flux limiter. Numerical examples are given to demonstrate the high-order accuracy and bound-preserving property of the numerical technique
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