April 19, 2014

April 20-22, 2009
Workshop on Computational Methods for Hyperbolic Problems
University of Waterloo, Department of Applied Mathematics



Yasunori Aoki, University of Waterloo
High-order Finite Volume Element Method for Laplace Young Capillary Equation

Our aim is to construct a fast converging numerical method for the Laplace-Young Capillary Equation (LYE), which is a scalar nonlinear Partial Differential Equation (PDE) of elliptic type. We have extended the Linear Finite Volume Element Method described by Scott, Sander and Norbury to a high-order method. For bounded solutions, the high-order FVEM leads to the expected improvement in numerical convergence order. However, the solution of the LYE is not necessarily bounded, and our standard method does not lead to improved order of convergence for these singular solutions. Numerical convergence studies are presented and avenues for further research are discussed.

Dale Connor, University of Waterloo
Applying WENO Ideas to Interpolation of Two-dimensional Curves

We propose a method for interpolating a two-dimensional curve using Euler spirals. The algorithm uses an adaptive stencil to prevent unwanted oscillations from appearing in the approximation. The interpolating curve is C1 in well-behaved regions, and is C in regions poorly approximated by a smooth curve. This allows the approximation to accurately model shapes which are smooth with the exception of specific points. The algorithm uses a lower order technique to approximate the tangents at each point, these tangents are required to define a unique Euler Spiral which is used as the interpolating segment connecting consecutive points. Since the interpolation basis consists of geometric shapes, it is well suited to interpolation of both one- and two-dimensional problems.

Georges Djoumna, University of Waterloo
The influence of numerical quadrature on the characteristic-based advection schemes

The advection equation remains difficult to solve numerically due to the fact that the transport process takes place along the characteristic lines and that the information comes from the past. A semi-Lagrangian temporal discretization of the total derivative is used to obtain the characteristic-Galerkin formulation. It is shown that tracking the characteristics backward from the quadrature nodes provides an accurate treatment of the advection terms. Numerical stability and analytical results quantifying the amount of artificial viscosity induced by the method are presented in the case of the one dimensional linear advection equation, based on the modified equation approach. Is the order of convergence in the absence of numerical integration unaltered by the effect of numerical integration? This question is under investigation, some preliminary results are analyzed. Key Words: advection equation; characteristic backward tracking; modified equation; finite elements; semi-Lagrangian methods.

Lucian Ivan, University of Toronto Institute for Aerospace Studies
Adaptive High-Order Central ENO Method for Hyperbolic Conservation Laws

High-order schemes are currently being actively pursued in an effort to reduce the cost of large-scale scientific computing applications. Moreover, for numerical simulations of physically complex flows having a wide range of spatial and temporal scales both high-order discretizations and adaptive mesh refinement (AMR) are often demanded. For hyperbolic conservation laws the challenge has been to achieve accurate discretizations while coping in a reliable and robust fashion with discontinuities and shocks. A high-order central essentially non-oscillatory (CENO) finite-volume procedure with adaptive mesh refinement is presented for the solution of hyperbolic systems of equations. The CENO spatial discretization is based on a hybrid solution reconstruction procedure that combines the unlimited high-order k-exact least-squares reconstruction technique of Barth based on a fixed central stencil with a monotonicity preserving limited piecewise linear least-squares reconstruction algorithm. Switching in the hybrid procedure is determined by a solution smoothness indicator that indicates whether or not the solution is resolved on the computational mesh. The limited reconstruction procedure is applied to computational cells with under-resolved and/or non-smooth solution content and the unlimited k-exact reconstruction scheme is used for cells in which the solution is fully resolved. An h-refinement criterion based on the solution smoothness indicator is defined and used to control refinement of the body-fitted multi-block AMR mesh.

James McDonald, University of Toronto
Realizable Hyperbolic Moment Closures for Gaseous Flows

First-Order Hyperbolic partial differential equations describing viscous heat-conducting gas behaviour, such as those resulting from
moment methods of gaskinetic theory, have several computational advantages over traditional fluid-dynamic equations, such as the
Navier-Stokes equations (which are partially elliptic in nature). For example, their first-order nature leads to the narrowest possible
stencil requirements and make computational solutions less sensitive to grid irregularities. One hierarchy of moment systems which seem to
have many desirable mathematical properties are those which assume the distribution function is always that which maximizes entropy
while describing a given set of moments. These maximum-entropy systems, however, suffer from a major drawback: there exist physically
realizable moment values for which a maximum-entropy distribution function does not exist. In these regions the moment equations
breakdown and become ill-posed. A remedy for this problem consisting of a perturbation to the assumed form of the distribution function
which expands the region of realizability of the closure to include all physically possible states is presented.

Ruibin Qin, University of Waterloo
A Ghost Cell Approach to Imposing Solid Wall Boundary Conditions on Cut Cells

We propose a new technique for implementation of reflective solid wall boundary conditions for solutions of one-dimensional Euler equations using the discontinuous Galerkin method. We consider a case where the cell adjacent to the solid wall is much smaller than regular interior cells. This results in a very small global time step when an explicit time integrator is used. Such situations frequently arise with embedded boundary methods. A technique for reconstruction of the solution on an extended ghost cell is proposed which allows us to use a regular time step defined by inner cells throughout the domain. Numerical examples demonstrate validity of the approach.

Scott Rostrup, University of Waterloo
Simulating the Shallow Water Equations on Accelerator Architectures

Though originally designed with video games and graphics, accelerator architectures such as the Cell Processor (PS3) and GPUs (Graphics Processing Unit) are beginning to be adopted by the scientific, financial, and engineering communities. They are highly parallel architectures suitable for a wide array of data-parallel computing tasks including the numerical methods suitable for
simulating hyperbolic partial differential equations. These architectures are explored as accelerators for an explicit wave based Godunov scheme applied to the shallow water equations.

Christopher Subich, University of Waterloo
High order moving mesh methods

High order spectral or pseudospectral methods have excellent convergence properties for differential equations with smooth solutions, but their accuracy deteriorates in the presence of sharp, local features -- most notably shocks. In contrast, moving finite difference methods allow for dynamic resolution of sharp features, but finite difference methods are inherently lower-order. This work unifies both approaches, creating a uniformly high-order moving mesh method to solve the viscous Burgers equation to high accuracy.

L.T. Tran, University of Utah
An Improved ICE Method for Compressible Flow Problems in Uintah

The Implicit Continuous-fluid Eulerian(ICE), a semi-implicit finite-volume solver, is used for simulating problems in multiphase flow which span a wide area of science and engineering. ICE is utilized by the C-SAFE code Uintah at the University of Utah to simulate explosions, fires and other fluid and fluid-structure interaction phenomena. The implementation of ICE in the CSAFE code Uintah invokes operator splitting in which the solution consists of a separate Lagrangian phase and Eulerian phase. The choices of operators in these phases effect the behavior of the ICE numerical solution. We discuss the implementation of ICE method for the numerical solutions to systems of conservation laws. The implementation of ICE used in Uintah is given in many papers by Kashiwa at Los Alamos and extended to solve multifield cases by Harman at Utah. In its original form the ICE algorithm does not perform as well as the best current methods for compressible flow problems. The poster to be presented shows how the ICE algorithm may be modified so that it provides good solutions for compressible high speed flow. This involves using limiters for calculating fluxing velocities at faces, and applying a gradient limiter
for approximating the advected quantities among cells. Computational results for a number of test problems are used to demonstrate the effectiveness of the approach.

Acknowledgement: This work was supported by the University of Utah's Center for the Simulation of Accidental Fires and Explosions (C-SAFE) and funded by the Department of Energy under subcontract No. B524196.

Joint work with M.Berzins