MS13: SIAM CSS student chapter presentations


Session A: Oct.1, 10:40am-12:00pm, Classroom Building 118
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 118

10:40am-11:00am (Oct 1)
CLB 118

Yifan Hu,
Iowa State University

Efficient Regionally-Implicit Discontinuous Galerkin Methods: A Dimensional Splitting Strategy for Linear Hyperbolic Systems


The regionally-implicit discontinuous Galerkin (RIDG) method is an extension of the prediction-correction formulation of the Lax-Wendroff discontinuous Galerkin method, where the prediction step is a regionally-implicit spacetime method that produces reconstructed solutions in spacetime, and the correction step is an explicit update based on those reconstructed solutions. In this work we developed a more efficient version of the RIDG scheme through a dimensional splitting strategy in the prediction step. We show that this new approach inherits the improved stability from the original RIDG method, and reduces the computational cost for multidimensional linear hyperbolic systems such as the advection equation and the wave equation.

11:00am-11:20am (Oct 1)
CLB 118
Ian Pelakh,
Iowa State University

A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods
  Nonlinear hyperbolic conservation laws admit singular solutions such as shockwaves (discontinuities in conserved variables), rarefaction waves (discontinuities in derivatives), and vacuum states (loss of strong hyperbolicity). When ostensibly high-order numerical methods are applied in such solution regimes, unphysical oscillations present themselves that can lead to large errors and a breakdown of the numerical simulation. In this work we develop a new Lax-Wendroff discontinuous Galerkin (LxW-DG) method with a limiting strategy that keeps the solution non-oscillatory and positivity-preserving for relevant variables, such as height in the shallow water equations and density and pressure in the compressible Euler equations. The proposed LxW-DG scheme updates the solution over each time-step with a locally-implicit predictor followed by an explicit corrector. The locally-implicit prediction phase is formulated in terms of primitive variables, which greatly simplifies the solver. The resulting system of nonlinear algebraic equations are approximately solved via a Picard iteration, where the number of iterations is equal to the order of accuracy of the method. The correction phase is an explicit evaluation formulated in terms of conservative variables in order to guarantee numerical conservation. In order to achieve full positivity-preservation, limiting is required in both the prediction and correction steps. The resulting scheme is applied to several standard test cases for the shallow water and compressible Euler equations. All of the presented examples are written in a freely available open-source Python code.

11:20am-11:40am (Oct 1)
CLB 118
James Burton,
University of Arkansas

Optimization of the Glimm’s scheme random choice method for mutliphasic flow simulations
  Numerical methods for solving systems of hyperbolic conservation laws must contend with shock formation and propagation. In particular, the Euler equations of compressible fluid dynamics require stable, accurate, and robust algorithms for shock computations. The method of choice for our simulation of compressible multiphase flows in 1D is the Glimm’s Scheme because of its good algorithmic properties, especially near discontinuities. Glimm’s Scheme using the random choice method (RCM) is revisited to investigate convergence properties using low-discrepancy sampling methods. A set of van der Corput sampling sequences is examined to determine the optimal choice in sampling sequence in five test cases. In each test case, simulations on various mesh sizes for each sequence are performed to determine the optimal sampling choice with a good convergence rate.

11:40am-12:00pm (Oct 1)
CLB 118
Ryan Holley,
University of Arkansas

Simulation of turbulent mixing due to Richtmyer-Meshkov instability using high order weighted essentially non-oscillatory schemes
  Turbulent mixing due to hydrodynamic instabilities occurs in a broad spectrum of engineering, astrophysical and geophysical applications. Theory, experiment, and numerical simulation help us to understand the dynamics of hydro-dynamically unstable interfaces between fluids. In our present simulations, higher order weighted essentially non-oscillatory (WENO) methods are used. We first introduce the WENO methods in solving hyperbolic partial differential equations. WENO schemes are high order accurate upwind finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The convex combinations of candidate stencils are chosen to approximate the flux at cell boundaries to a high order of accuracy and avoid oscillations near shocks. Lastly, the use of higher order WENO methods is investigated in the simulation of turbulent mixing due to Richtmyer-Meshkov Instability.

3:10pm-3:30pm (Oct 1)
CLB 118
Preeti Sar,
Iowa State University

Asymptotic-preserving schemes for the kinetic Boltzmann-BGK equation
  In this talk I will review two important asymptotic-preserving (AP) schemes for solving the Boltzmann-BGK equation. The Boltzmann equation describes this motion of a fluid for moderate to large Knudsen numbers. Numerically, the BGK model approximates the Boltzmann equation for moderate Knudsen numbers. The paper “Implicit-Explicit Schemes for BGK Equation” (Pierraccini and Puppo) presents a new class of numerical methods for the BGK model of kinetic equations which is based on an implicit-explicit time discretization. The small scales in kinetic and hyperbolic equations lead to different asymptotic regimes which are expensive to solve numerically. Asymptotic preserving schemes are efficient in these regimes, which preserve at the discrete level, the asymptotic limit which drives the microscopic equation to its macroscopic equation. The paper “Uniformly Stable Numerical Schemes for the Boltzmann Equation preserving the Compressible Navier-Stokes Asymptotics” (Bennoune, Lemou and Mieussens) develops a numerical method to solve Boltzmann like equations of kinetic theory which can capture the compressible Navier-Stokes dynamics at small Knudsen numbers. This method, which is based on the micro/macro decomposition technique, is performed in all the phase space and leads to an equivalent formulation of the Boltzmann equation that couples a kinetic equation with macroscopic ones.

3:30pm-3:50pm (Oct 1)
CLB 118
Xuan Gu,
University of Arkansas

Model order reduction for elliptic partial differential equations
  The blooming strategies of finding numerical solutions to partial differential equations (PDE) have been developed in this century, while the computational complexity still poses a major obstacle to large scale PDEs. In this talk, we present multiple reduced order models (ROMs) with the finite element solver on the elliptic second order PDE which is the generalized Laplace equation. The PDEs with specific boundary conditions are firstly discretized and parameterized into full order linear finite element system using NumPy/SciPy packages. The full order model is projected onto the reduced space which is spanned by the reduced basis. We present the construction of reduced basis using strong greedy algorithm, proper orthogonal decomposition, and weak greedy algorithm. A coupled computation framework is utilized with the discretization and the reduced basis construction implemented in the open source pyMOR package, integrated into the finite element solver package deal.II. Finally, we demonstrate numerical solutions and error analysis on elliptic equations to illustrate the efficiency and accuracy of the ROMs.

3:50pm-4:10pm (Oct 1)
CLB 118
Tyler Kroells,
Iowa State University

Cell-Average Based Neural Network Solvers For Partial Differential Equation
  In this talk, we will develop the Cell Average Neural Network (CANN) method to solve time dependent partial differential equations, a method motivated by finite volume schemes. The CANN is based on the weak or integral formula for PDE’s. We use a simple feed forward network to learn the difference between cell averages in consecutive time steps. Training data is generated from one solution trajectory, and once the network is trained it operates similar to an explicit one step finite volume method. Unlike a traditional finite volume method, the CANN method can be adapted to larger time step sizes, outside of CFL restrictions. A large time step such as dt=O(dx) can be applied to evolve the solution forward in time. We will focus on numerical examples for the advection equation and heat equation, but the method has also been successfully applied to other, more complex equations.