MS07: New trends on discontinuous Galerkin methods for partial differential equations


Session A: Oct.1, 10:40am-12:00pm, Classroom Building 106
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 106
Session C: Oct.1, 5:00pm-6:20pm, Classroom Building 106

10:40am-11:00am (Oct 1)
CLB 106

Yang Yang,
Michigan Technological University

A reinterpreted discrete fracture model for fracture and barrier networks


In this talk, we construct the reinterpreted discrete fracture model for flow simulation of fractured porous media containing flow blocking barriers on non-conforming meshes. The methodology of the approach is to modify the traditional Darcy’s law into the hybrid-dimensional Darcy’s law where fractures and barriers are represented as Dirac-delta functions contained in the permeability tensor and resistance tensor, respectively. This model is able to account for the influence of both highly conductive fractures and blocking barriers accurately on non-conforming meshes. The local discontinuous Galerkin (LDG) method is employed to accommodate the form of the hybrid-dimensional Darcy’s law and the nature of the pressure/flux discontinuity. The performance of the model is demonstrated by several numerical tests.

11:00am-11:20am (Oct 1)
CLB 106
Yue Kang,
Michigan Technological University
Bound-preserving discontinuous Galerkin methods with second-order implicit pressure explicit concentration time marching for compressible miscible displacements in porous media
  In this talk,we will introduce the construction of the bound-preserving interior penalty discontinuous Galerkin(IPDG) methods with a second-order implicit pressure explicit concentration(SIPEC) time marching for the coupled system of two-component compressible miscible displacements. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta(SSP-RK2) method. The main idea is to treat the pressure equation implicitly and the concentration equation explicitly. However, this treatment would result in a first-order accurate scheme. We can propose a correction stage to compensate for the second-order accuracy in each time step. Numerical experiments will be given to demonstrate that the proposed scheme can reduce the computational cost significantly compared with explicit schemes.

11:20am-11:40am (Oct 1)
CLB 106
Fangyao Zhu,
Michigan Technological University
Bound-preserving discontinuous Galerkin Methods with Patankar time discretization for chemical reacting flows
  I will talk about bound-preserving DG methods for chemical reactive flows. For this problem we must ensure the density and internal energy are kept positive, and the mass fraction of each species is between 0 and 1. We apply the bound-preserving technique to the DG methods. Though traditional positivity-preserving techniques can successfully yield positive density, internal energy, and mass fractions, it may not enforce the upper bound 1 of the mass fractions. To solve this problem, we need to make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density; choose conservative time integrations such that the summation of the mass fractions is preserved. With the above two conditions, the positive mass fractions have summation 1, then they are all between 0 and 1. For time discretization, we apply the modified Runge-Kutta/multi- step Patankar methods. Such methods can handle stiff sources with relatively large time steps, preserve the positivity of the target variables, and keep the summation of the mass fractions to be 1. To evolve in time, suitable slope limiters can be applied to enforce the positivity of the solutions. Numerical examples will be shown.

11:40am-12:00pm (Oct 1)
CLB 106

Yuan Liu,
Wichita State University

Discontinuous Galerkin methods for network patterning phase-field models
  In this talk, we will discuss a class of discontinuous Galerkin methods under the scalar auxiliary variable framework (SAV-DG) to solve a biological patterning model in the form of parabolic-elliptic PDE system. In particular, mixed-type discontinuous Galerkin approximations are used for the spatial discretization, aiming to achieve the balance the high resolution and computational cost. Second and third order backward differentiation formula are considered under SAV framework for energy stability. Numerical experiments are provided to show the effectiveness of the fully discrete schemes and the governing factors of patterning formation.

3:10pm-3:30pm (Oct 1)
CLB 106
Thomas Lewis,
The University of North Carolina at Greensboro

Dual‐Wind Discontinuous Galerkin Methods for Fully Nonlinear Second Order PDEs
  We will introduce a class of dual-wind discontinuous Galerkin (DG) methods for approximating viscosity solutions of fully nonlinear second order elliptic partial differential equations. The methods will be motivated using the DG interior calculus to define discrete upwind and downwind gradient operators as well as symmetric operators for directly discretizing the Hessian operator in the PDE. Admissibility and stability results will be discussed. By using the dual-wind methodology, the methods are stable without requiring a jump penalization term. The DG methods are a natural extension of narrow-stencil, generalized monotone finite difference methods for approximating viscosity solutions of fully nonlinear PDEs.

3:30pm-3:50pm (Oct 1)
CLB 106
Aaron Rapp,
University of the Virgin Islands

Dual-Wind Discontinuous Galerkin Methods for Time-Dependent Hamilton Jacobi Equation
  This talk will discuss a new family of dual-wind discontinuous Galerkin (DG) methods for Hamilton-Jacobi equations (HJEs) and their vanishing viscosity regularizations. The proposed methods, which are non-monotone, utilize a dual-winding methodology and a new skew-symmetric DG derivative operator that, when combined, eliminate the need for choosing indeterminable penalty constants. Admissibility and stability are established for the proposed dual-wind DG methods on stationary HJEs. The stability results for stationary HJEs are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes, and are believed to still hold for time-dependent HJEs. Many numerical experiments are provided to gauge the performance of the new DWDG methods with time stepping methods on time-dependent HJEs.

3:50pm-4:10pm (Oct 1)
CLB 106
Yangwen Zhang,
Carnegie Mellon University

Superconvergent interpolatory HDG methods for nonlinear reaction diffusion equations II: HHO-inspired methods
  In [J. Sci. Comput., 81:2188-2212, 2019], we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction-diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time intergration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree k>=1. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in [ESAIM Math. Model. Numer. Anal., 50(3):635650, 2016] and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein. We prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for k>=0 by some methods. We present numerical results to illustrate the convergence theory.

4:10pm-4:30pm (Oct 1)
CLB 106
Mustafa Danis,
Iowa State University

A generalized framework for direct discontinuous Galerkin methods
  In this talk, we present a unified, generalized framework for the direct discontinuous Galerkin (DDG) methods. Unlike the original DDG methods, the antiderivative of the nonlinear diffusion matrix is not needed in the new framework. This leads to a considerable simplification in the numerical flux formulation such that the standard DDG numerical flux for the heat equation can be used for general nonlinear diffusion equations without further modifications. We also present the nonlinear stability analysis of the new DDG methods and their extension to the more general system of conservation laws. We also consider the application of the new DDG methods to various nonlinear diffusion equations, including the compressible Navier-Stokes in subsonic and hypersonic flow settings. In the numerical experiments, we demonstrate that the interface correction and symmetric DDG versions achieve optimal convergence and are superior to the nonsymmetric DDG. Singular or blow-up solutions are also well captured with the new DDG methods.

5:00pm-5:20pm (Oct 1)
CLB 106
Wei Guo,
Texas Tech University

A Local Macroscopic Conservative (LoMaC) low rank tensor method with the discontinuous Galerkin method for the Vlasov dynamics
  In this talk, we present a novel Local Macroscopic Conservative (LoMaC) low rank tensor method with discontinuous Galerkin (DG) discretization for the physical and phase spaces for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. The LoMaC low rank tensor algorithm (recently developed in arXiv:2207.00518) simultaneously evolves the macroscopic conservation laws of mass, momentum and energy using the kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables. This work is a generalization of our previous development, but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term. The algorithm is developed in a similar fashion as that for a finite difference scheme, by observing that the DG method can be viewed equivalently in a nodal fashion. With the nodal DG method, assuming a tensorized computational grid, one will be able to (1) derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms, and (2) define a weighted inner product space based on the nodal DG grid points. The algorithm can be extended to the high dimensional problems by hierarchical Tucker decomposition of solution tensors and a corresponding conservative projection algorithm. In a similar spirit, the algorithm can be extended to DG methods on nodal points of an unstructured mesh, or to other types of discretization, e.g. the spectral method in velocity direction. Extensive numerical results are performed to showcase the efficacy of the method.

5:20pm-5:40pm (Oct 1)
CLB 106
James Rossmanith,
Iowa State University

Locally-implicit discontinuous Galerkin schemes for the kinetic Boltzmann-BGK system that are arbitrarily high-order and asymptotic-preserving
  The kinetic Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collision operator allows for the simulation of gas dynamics over a wide range of Knudsen numbers with a simplified collision operator. Efficient numerical methods for Boltzmann-BGK should be asymptotic-preserving, which allows the numerical method to be stable at fixed mesh parameters for any value of the Knudsen number, including in the fluid (very small Knudsen numbers), slip flow (small Knudsen numbers), transition (moderate Knudsen numbers), and free molecular flow (large Knudsen numbers) regimes. In this work we develop a novel approach for solving the Boltzmann-BGK equation for achieving both arbitrary high-order and the asymptotic-preserving property. The proposed method is a locally-implicit discontinuous Galerkin (LIDG) scheme with careful modification in both the prediction and correction steps to achieve the asymptotic-preserving property. Some key advantages of the proposed schemes are: (1) no splitting between macroscale and microscale components of the distribution function is required; (2) only a single unified time-discretization is required; and (3) arbitrary high-order in both space and time can be achieved simply by increasing the spatial polynomial order in each element. Several numerical examples are shown to demonstrate the effectiveness of the proposed numerical scheme.

5:40pm-6:00pm (Oct 1)
CLB 106
Jeonghun Lee,
Baylor University

Hybridizable discontinuous Galerkin methods for coupled Stokes-Biot problems
  In this work we develop hybridizable discontinuous Galerkin (HDG) methods for problems such that the Stokes equations and the Biot consolidation equations are coupled with interface. In our HDG methods the compressibilities of fluid and poroelastic matrix, and the fluid mass in poroelastic domain are strongly conservative. We show a priori error estimates of numerical solutions which are robust in the sense that the constants of error bounds are uniformly bounded for nearly incompressible materials and do not grow exponentially in time. This is a joint work with Aycil Cesmelioglu at Oakland University and Sander Rhebergen at University of Waterloo.

6:00pm-6:20pm (Oct 1)
CLB 106
Chen Liu,
Purdue University

A conservative and positivity-preserving implicit-explicit approach for compressible fluid flow simulation
  In many demanding gas dynamics applications such as hypersonic ow simulation, the compressible Navier-Stokes (NS) equations form one of the most popular and important models. In this talk, we proposed an implicit-explicit approach for solving compressible NS system. We utilize operator splitting technique separates the compressible NS equations into a hyperbolic sub-problem and a parabolic sub-problem. The discontinuous Galerkin discretization of hyperbolic sub-problem is invariant domain preserving. The positivity-preserving property for the parabolic sub-problem is constructed via the monotonicity of the system matrix. Our proposed algorithm enjoys the property that the CFL condition is independent of the Reynolds number. Therefore, they are highly preferred and suitable for simulating realistic physical and engineering problems.