Notes on the Discontinuous Galerkin methods for the numerical simulation of hyperbolic equations 1 General Context 1.1 Bibliography

The roots of Discontinuous Galerkin (DG) methods is usually attributed to Reed and Hills in a paper published in 1973 on the numerical approximation of the neutron transport equation [18]. In fact, the adventure really started with a rather thoroughfull series of five papers by Cockburn and Shu in the late 80's [7, 5, 9, 6, 8]. Then, the fame of the method, which could be seen as a compromise between Finite Elements (the center of the method being a weak formulation) and Finite Volumes (the basis functions are defined cell-wise, the cells being the elements of the primal mesh) increased and slowly investigated successfully all the domains of Partial Differential Equations numerical integration. In particular, one can cite the ground papers for the common treatment of convection-diffusion equations [4, 3] or the treatment of pure elliptic equations [2, 17]. For more information on the history of Discontinuous Galerkin method, please refer to section 1.1 of [15]. Today, DG methods are widely used in all kind of manners and have applications in almost all fields of applied mathematics. (TODO: cite applications and structured/unstructured meshes, steady/unsteady, etc...). The methods is now mature enough to deserve entire text books, among which I cite a reference book on Nodal DG Methods by Henthaven and Warburton [15] with the ground basis of DG integration, numerical analysis of its linear behavior and generalization to multiple dimensions. Lately, since 2010, thanks to a ground work of Zhang and Shu [26, 27, 25, 28, 29], Discontinuous Galerkin methods are eventually able to combine high order accuracy and certain preservation of convex constraints, such as the positivity of a given quantity, for example. These new steps forward are very promising since it brings us very close to the "Ultimate Conservative Scheme", [23, 1]

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