Local Discontinuous Galerkin Methods for Solving Convection-Diffusion on Surface

Zhiliang Xu

University of Notre Dame

In this talk, we present a LDG method for solving convection-diffusion and Cahn-Hilliard equations on surfaces, respectively. Piecewise linear triangles are used to approximate the surface. Because of this, the scheme is 2nd-order accurate. Stability analysis was performed to show that the scheme is energy stable.

Development of Sparse Grid DG Methods

Yingda Cheng

Michigan State University

In this talk, we present the construction of sparse grid discontinuous Galerkin methods. We will consider some recent developments in the applications of kinetic equations and other type of nonlinear problems.

Lax-Wendroff DG schemes for Quasi-exponential Moment-closures

James Rossmanith

Iowa State University

In many applications the dynamics of gas and plasma can be accurately modeled using kinetic Boltzmann equations. These equations are integro-differential systems posed in a high-dimensional phase space, which is typically comprised of the spatial coordinates and the velocity coordinates. If the system is sufficiently collisional, the kinetic equations may be replaced by a fluid approximation that is posed in physical space (i.e., a lower dimensional space than the full phase space). The precise form of the fluid approximation depends on the choice of the moment-closure. In general, finding a suitable robust moment-closure is still an open scientific problem.

In this work we consider a specific moment-closure based on a nonextensive entropy formulation. In particular, the true distribution is replaced by a Maxwellian distribution multiplied by a quasi-exponential function. We develop a high-order, locally-implicit, discontinuous Galerkin scheme to numerically solve resulting fluid equations. The numerical update is broken into two parts:

(1) an update for the background Maxwellian distribution, and

(2) an update for the non-Maxwellian corrections.

We also develop limiters that guarantee that the inversion problem between moments of the distribution function and the parameters in the quasi-exponential function is well-posed.

Discontinuous Galerkin Methods for solving a Thin-Film Equation

Caleb Logemann

Iowa State University

The particular water-mass transport equation that we consider is the thin-film equation. The thin-film equation results from an asymptotic limit of the Navier-Stokes equations and describes how a thin-film of water flows over a surface. The thin-film equation contains both a convection and diffusion term and thus is most often handled with operator splitting. We describe how to use an explicit Runge-Kutta DG method for solving the convection equation and an implicit finite difference method for solving the diffusion equation.

High-order Bound-preserving Discontinuous Galerkin Methods for Stiff Multispecies Detonation

Yang Yang

Michigan Technological University

In this talk, we develop high-order bound-preserving discontinuous Galerkin (DG) methods for multispecies and multireaction chemical reactive flows. In this problem, density and pressure are nonnegative, and the mass fraction for each species, should be between 0 and 1, where M is the total number of species. There are three main difficulties. First of all, most of the previous bound-preserving techniques were based on Euler forward time discretization. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum-principle, and most of the previous techniques cannot be applied. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on the second-order Strang splitting methods where the flux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. In this paper, we will solve all the three problems given above. The high-order time integration does not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization is explicit and can handle the stiff source with large time step. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.

Third Order Positivity-Preserving Direct Discontinuous Galerkin Method for Keller-Segel Chemotaxis Equations

Jue Yan

Iowa State University

We develop a new direct discontinuous Galerkin (DDG) method to solve Keller-Segel Chemotaxis equations. One unique feature of our method is that we introduce no extra variables to approximate the gradient of the chemical concentration and solve the system directly with DDG method. We obtain optimal (k+1)th order convergence with kth degree piecewise polynomials approximations, even on random none uniform meshes. Furthermore, we prove the cell density solution is maintained positive at all time levels with at least third order of accuracy. Cell density blow up phenomena is captured well.

A Mixed Discontinuous Galerkin Method without Interior Penalty for Time-dependent Fourth Order Problems

Peimeng Yin

Iowa State University

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are $L^2$ stable even without interior penalty. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal $L^2$ error estimate of $O(h^{k+1})$ for polynomials of degree $k$ for semi-discrete DG schemes, and the $L^2$ error of $O(h^{k+1} +(Δt)^2)$ for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes.

An Adaptive Local Discontinuous Galerkin Method for Nonlinear Two-point Boundary-value Problems

Mahboub Baccouch

University of Nebraska at Omaha

In this talk, we present an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs). We prove optimal $L^2$ error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be $p+1$, when piecewise polynomials of degree at most $p$ are used. We further prove that the LDG solutions are superconvergent with order $p+2$ toward Gauss-Radau projections of the exact solutions. We then use the superconvergence results to show that the significant parts of the local discretization errors are proportional to $(p+1)$-degree Radau polynomials. These results allow us to construct a residual-based a posteriori error estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed a posteriori error estimates converge to the exact errors in the $L^2$-norm with order of convergence $p+2$. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.