24 research outputs found
HP Primal Discontinuous Galerkin Finite Element Methods for Two-Phase Flow in Porous Media
The understanding and modeling of multiphase flow has been a challenging research problem for many years. Among the important applications of the two-phase flow problem are simulation of the oil recovery and environmental protection. The two-phase flow problem in porous media is mathematically modeled by a nonlinear system of coupled partial differential equations that express the conservation laws of mass and momentum. In general, these equations can only be solved by the use of numerical methods.The research in the thesis mainly focuses on the numerical simulation and analysis of different models of incompressible two-phase flow in porous media using primal Discontinuous Galerkin (DG) finite element methods.First, in our work we derive sharp computable lower bounds of the penalty parametersfor stable and convergent symmetric interior penalty Galerkin methods (SIPG) applied to the elliptic problem. In particular, we obtain the explicit dependence of the coercivity constants with respect to the polynomial degrees and the angles of the mesh elements. These bounds play an important role in the derivation of the stability bounds for the SIPG method applied to the the two-phase flow problem. Next, we consider three different implicit pressure-saturation formulations for two-phase flow. We study both h- and p-versions, i.e. convergence is obtained by either refining the mesh or by increasing the polynomial degree. We develop numerical analysis for one of the pressure-saturation formulations. Numerical tests which confirm our theoretical results are presented. Some validation of the proposed schemes, comparison between numerical solutions which are obtained by different schemes and numerical simulations of benchmark problems also given
Energy Stable and Structure-Preserving Schemes for the Stochastic Galerkin Shallow Water Equations
The shallow water flow model is widely used to describe water flows in
rivers, lakes, and coastal areas. Accounting for uncertainty in the
corresponding transport-dominated nonlinear PDE models presents theoretical and
numerical challenges that motivate the central advances of this paper. Starting
with a spatially one-dimensional hyperbolicity-preserving,
positivity-preserving stochastic Galerkin formulation of the
parametric/uncertain shallow water equations, we derive an entropy-entropy flux
pair for the system. We exploit this entropy-entropy flux pair to construct
structure-preserving second-order energy conservative, and first- and
second-order energy stable finite volume schemes for the stochastic Galerkin
shallow water system. The performance of the methods is illustrated on several
numerical experiments
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that
the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied
to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its
high resolution and robustness in a number of numerical examples