Course Project Presentations

Students will present Course Projects in two sessions:

Wednesday, March 11, 4:00 - 5:00pm

In JHN 026

4:00 - 4:20: Krithika Manohar and Tommaso Buvoli

• TITLE: Introduction to WENO Methods
• ABSTRACT: Essentially non-oscillatory or ENO methods have historically proven successful at capturing shock discontinuities by choosing the smoothest interpolating polynomial at several neighboring stencils. Weighted ENO or WENO methods use a convex combination of the interpolating polynomial at all stencils and hence preserve both the high-resolution for shocks and high-order accuracy for smooth data. We implement and test these schemes on one-dimensional hyperbolic equations such as the advection and burgers equations with shocks. We will also include comparisons with CLAWPACK’s results for non-WENO schemes. We also plan to investigate the benefits of using Total Variation Diminishing (TVD) Runge-Kutta schemes versus traditional non-TVD integrators.
• MATERIALS: https://github.com/amath574w2015/am574-group05

4:20 - 4:40: Qi Guo and Peng Zheng

• TITLE: Models of Traffic Flow with Discontinuous and Non-convex Flux
• ABSTRACT: In this project, we investigate two models of traffic flow based on Lighthill-Whitham-Richards model. In the first part, we look into the model of traffic flow on freeway, where the flux function is discontinuous and piecewise linear. We utilize the method of mollification to smooth out the discontinuity, and construct the convex hull to solve the problem. Additionally, a numerical PDE solver in CLAWPACK and a ODE solver of car-following model are designed to simulate the results. In the second part, the car-following model of night-time driving is explored. With or without perturbing the velocities, we could observe the instability and clustering of cars with uniform initial density.
• MATERIALS: https://github.com/amath574w2015/am574-group07

4:40 - 5:00: Kelsey Maass and Brisa Davis

• TITLE: Comparison of Two Second Order Traffic Flow Models
• ABSTRACT: While the Lighthill-Whitham-Richards traffic flow model behaves well macroscopically, it does not accurately describe how vehicles travel through shocks. We compare two second order models, the Payne-Whitman model and the Aw-Rascle model, that attempt to improve the first order LWR model. Specifically, we demonstrate that the PW model’s representation of traffic as a fluid ignores the anisotropic nature of cars, which leads to unrealistic results. Next we consider the AR model, which utilizes a convective derivative to resolve the problem of negative velocities present in the PW model. Through this talk we hope to highlight the fact that introducing higher order relations does not automatically improve the accuracy of modeling a given physical system, illustrating the importance of validation in modeling.
• MATERIALS: https://github.com/amath574w2015/am574-group06

Friday, March 13, 3:30 - 5:30pm

In OUG 141 (Odegaard Library)

3:30 - 3:50: Saumya Sinha and Kenneth Roche

• TITLE: Adaptive Mesh Refinement for 1D Hyperbolic PDEs
• ABSTRACT: We describe an adaptive mesh refinement algorithm that extends high resolution wave-progpagation techniques to hyperbolic systems in non-conservative form. The algorithm was implemented and tested for simple 1D problems. Results are compared to static mesh solutions for the same problems.
• MATERIALS: https://github.com/amath574w2015/am574-group03

3:50 - 4:10: Devin Light and Scott Moe

• TITLE: A p-Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws in 1D
• ABSTRACT: Discontinuous Galerkin (DG) methods are becoming increasingly popular tools for the numerical integration of hyperbolic conservation laws. DG methods provide a natural extension of finite volume methods to higher orders while maintaining a compact stencil and exhibit a number of desirable computational features. However, as problems of larger and larger scopes are considered it is increasingly necessary to implement an adaptive method which devotes the finite degrees of freedom to where they are most needed in the domain. To that end we propose a novel p-adaptive DG scheme which uses a hierarchical basis and allows the degree of the local polynomial approximation to vary between cells. This method either adds or removes degrees of freedom for the next step in the integration based on the behavior of the coefficient of the highest degree polynomial basis present in the current approximation. The efficiency and accuracy performance of the proposed method will be measured against a non-adapting scheme on several standard tests.
• MATERIALS: https://github.com/amath574w2015/am574-group02

4:10 - 4:30: Hai Zhu and Xin Yang

• TITLE: The f-wave method for nonlinear conservation laws with spatially varying flux
• ABSTRACT: The wave-propagation form has been shown to be a nice way to implement the finite volume method for hyperbolic conservation laws. In this project we study one generalization of the standard wave-propagation method which decomposes the flux into waves of the eigenvectors of the Jacobian matrix rather than decomposing the conserved quantities. Computational experiments using the f-wave method are performed for 1D heterogeneous nonlinear elastic wave model.
• MATERIALS: https://github.com/amath574w2015/am574-group01

4:30 - 4:50: Chris Uyeda and Alex Li

• TITLE: Convergence of Several Fixed Geometry Nozzles Using the Pseudo-1D Euler Equations
• ABSTRACT: The pseudo-1D Euler equations will be used to simulate several different converging-diverging nozzle geometries and test each nozzles’ ability to converge to a steady state with a shock in the diverging section of the nozzle. Each simulation will use a fixed pressure ratio starting from a pressure reservoir and exhausting to the ambient environment. The location of the shock from simulations will be compared to the analytical quasi-1D solution derived from isentropic and shock relations in the nozzle. In addition to tracking the shock location, the flow property distribution will also be analyzed to ensure the expected physics of flow in a converging-diverging nozzle is satisfied.
• MATERIALS: https://github.com/amath574w2015/am574-group04

4:50 - 5:10: Jacob Ortega-Gingrich and Chen Xin

• TITLE: Augmented approximate Riemann solvers for the shallow water equations with variable bathymetry
• ABSTRACT: We describe an augmented Riemann solver for the one-dimensional shallow water equations with variable topography suggested by David George which addresses a number of the needs of applications involving flooding and small perturbations of delicate steady states. Fluid flows over varying topography, for example add a source term which the numerical method must balance with jumps in momentum and depth in order to preserve delicate steady states, such as an ocean at rest. Furthermore, applications involving flooding, such as the modeling of tsunami inundation, require a Riemann solver that can handle dry cells and preserve depth non-negativity. The augmented solver herein discussed satisfies these properties by amalgamating various aspects of existing solvers such as the HLLE solvers and f-wave approaches and the addition of a stationary steady state wave to account for the source term. Additionally, we discuss some specific techniques which may be use to handle various challenging situations which may arise in a model such as the handling of steep shorelines. Finally, we discuss an implementation of this augmented Riemann solver for the one-dimensional shallow water equations and present a few numerical demonstrations.
• MATERIALS: https://github.com/amath574w2015/am574-group09

5:10 - 5:30: Kaspar Mueller and Shawn Qin

• TITLE: Hydraulic bore interaction with a column - A comparison between the solution of the shallow equation and experimental results
• ABSTRACT: In this paper we compare the solution of the shallow water equations with experimental results. The experiment simulates the interaction between an incident bore and a free-standing coastal structure. The shallow water equations are solved using the GeoClaw solver of the CLAWPACK software. Three different cases are simulated and compared. In the first case a simple dam break problem is performed and evaluated. The history of the wave height and velocity at the center location, where the column will be mounted are compared. In the second case, a square column is added and wave height at various locations in front of the column are measured and compared. For the third case, the square column is replaced by a cylinder column and the same variables are measured and compared. The discrepancy between the GeoClaw simulation and the experimental results are discussed and analyzed.
• MATERIALS: https://github.com/amath574w2015/am574-group08

5:30 – 6:30: Celebration with munchies