In this talk, we develop third-order conservative sign-preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows. In this problem, some unknown variables has special physical bounds, for example, the mass fraction for the ith species, denoted as z_i, 1<=i<=M, should be between 0 and 1, where M is the total number of species. We would like to construct suitable numerical techniques to preserve those physical bounds. There are four main difficulties in constructing high-order bound-preserving techniques. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum-principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on 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. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this talk, we will construct third-order conservative sign-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting. Moreover, the time discretization 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.

In this talk, we propose a semi-Lagrangian discontinuous Galerkin method coupled with commutator-free exponential integrators for nonlinear transport problems. The commutator-free exponential integrators were proposed by Celledoni, et al. (FCGS 19, 341-352,2003), which can be viewed as the method that decomposes the transport equation into a composition of linearized transport equations. The resulting linearized transport equations are solved by the semi-Lagrangian discontinuous Galerkin method proposed in Cai, et al. (J. Sci. Comput. 514-542, 2017). In the end, the overall scheme can enjoy advantages such as up to third order accuracy in both space and time discretization, allowing for large time-stepping size, mass conservative, no splitting error, positivity-preserving and ability in resolving complex solution structures. All these advantages can be verified by applying the schemes to solve the Vlasov-Poisson systems and the Guiding center Vlasov model.

The simulation of blood flow in arteries can be modeled by a system of conservation laws and have a range of applications in medical contexts. This system of partial differential equations is in the same vein as the shallow water equations. We present well-balanced discontinuous Galerkin methods for the blood flow model which preserve the general moving equilibrium. Schemes for systems with zero-velocity have been recently been addressed, however we focus on the development of schemes that consider general moving equilibrium. Recovery of well-balanced states via appropriate source term approximations and approximations of the numerical fluxes are the key ideas. Numerical examples will be presented to verify the well-balanced property, high order accuracy, and good resolution for both smooth and discontinuous solutions.

We propose a semi-Lagrangian (SL) discontinuous Galerkin (DG) method for the BGK model. With the SL method for transport and implicit discretization of the stiff relaxation term, the time stepping size can be relaxed and much larger than that from an Eulerian framework with explicit treatment of the source term. We perform accuracy analysis of the scheme both in the kinetic and the fluid regime. Stability property of the employed DIRK methods is also verified for linear kinetic transport equations in a diffusive scaling setting. Extensive numerical tests are performed to verify the high order accuracy of the proposed method for both consistent and inconsistent initial data, and for its performance on problems with shocks.

We develop a high order positivity-preserving limiter with interior penalty discontinuous Galerkin (IPDG) method for one-dimensional compressible Navier-Stokes equations. The density and pressure approximations are proved to be nonnegative at all time levels with uniform high order of accuracy. The limiter applies a CFL condition on the first-order forward Euler time discretization which makes the algorithm easy to implement and accessible to strong stability preserving (SSP) high order time discretizations. The simplicity of the formulation allows extension of this limiter to higher dimensions, and also to other DG schemes such as Direct DG (DDG).

The standard HDG methods (use degree k polynomial spaces for all variables) can lose optimal convergence in some cases, and one way to recover is to use a variant (which we call HDG+ for simplicity), where the degree of polynomials approximating the primal unknown is increased by 1, and the numerical flux is redefined with a reduced stabilization function. The existing analysis of HDG+ is quite different from the projection-based analysis of the standard HDG methods, where specific projections are defined to make the analysis simple and concise. We show that with some new analytical tools, we can similarly devise projections for HDG+ methods. This enables us to recycle many existing analyses of the standard HDG methods for the analyses of HDG+, and help understand better the connections between these several variants.

We present a new technique for computing highly accurate approximations to linear functionals obtained in terms of Galerkin approximations. This technique is devised specifically for numerical methods satisfying a Galerkin orthogonality property. We illustrate the technique on a simple model problem of approximating J(u), where J is a smooth functional and u is the solution of a Poisson problem. Our numerical results show that we can double the convergence order by only doubling the computational effort. Numerical results for other types of functionals are also presented.

The discrete ordinates discontinuous Galerkin (SN-DG) method is a well-established and practical approach for solving the radiative transport equation. In this paper, we study a low-memory variation of the upwind SN-DG method. The proposed method uses a smaller finite element space that is constructed by coupling spatial unknowns across collocation angles, thereby yielding an approximation with fewer degrees of freedom than the standard method. Like the original SN-DG method, the low memory variation still preserves the asymptotic diffusion limit and maintains the characteristic structure needed for mesh sweeping algorithms. While we observe second-order convergence in scattering dominated, diffusive regime, the low-memory method is in general only first-order accurate. To address this issue, we use upwind reconstruction to recover second-order accuracy. For both methods, numerical procedures based on upwind sweeps are proposed to reduce the system dimension in the underlying Krylov solver strategy.