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Research has been performed on weighted essentially non-oscillatory schemes and discontinuous Galerkin methods, and other related numerical methods, which are high order accurate numerical methods for solving problems with shocks and other complicated solution structures. New algorithm aspects include subcell resolution for non-conservative systems, high order well balanced schemes, stable Lagrangian schemes, schemes for front propagation with obstacles, and homotopy method for steady states. Applications include high order simulations for 3D gaseous detonations, sound generation study via vortex interactions, turbulence simulations, simulations of resonant photons, and dynamic continuum models for traffic flows in urban cities with efficient and stable numerical simulations.

We have performed research on the algorithm development, analysis, implementation and application for high order weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and related methods, for solving computational fluid dynamics (CFD) problems and other applications containing strong shock waves and complicated smooth region structures. The goal of this research is to help obtaining more robust, cost effective, and reliable numerical tools for computational fluid dynamics problems and other applications, that are of interest to AFOSR. 22 papers in positivity-preserving high order schemes, stable and accurate boundary conditions for high order finite difference methods, efficient semi-Lagrangian finite difference schemes and other topics are published in refereed journals.

We investigate the angular distribution of Ly(alpha) photons transferring in or emergent from an optically thick medium. Since the evolutions of specific intensity I in the frequency space and the angular space are coupled from each other, we first develop the WENO numerical solver in order to find the time-dependent solutions of the integro-differential equation of I in the space of frequency and angular simultaneously. We show first that the solutions with the Eddington approximation, which assume I to be linearly dependent on the angular variable mu, yield similar frequency profiles of the photon flux as that without the Eddington approximation. However, the solutions of the mu distribution evolution are significantly different from that given by Eddington approximation. First, the angular distribution of I are found to be substantially dependent on the frequency of photons. For photons with the resonant frequency nu0, I contains only a linear term of mu;. For photons with frequency at the double peaks of the flux, the mu-distribution is highly anisotropic, in which most photons are in the direction of radial forward. Moreover, either at nu0 or at the double peaks, the mu distributions actually are independent of the initial mu distribution of photons of the source. This is because the photons with frequency either of nu0 or of the double peaks have undergone the process of forgetting their initial conditions due to the resonant scattering. We also show that the optically thick medium is a collimator of photons at the double peaks. Photons from the double peaks form a forward beam with very small spread angle.

Through direct numerical simulation (DNS) for the sound generated by the interaction between two isentropic vortices, it is found that the interaction between two vortices with a large difference in their strengths or scales can generate continuous strong noise. The interaction between two isentropic vortices results in the formation of two vortex dipoles, with each vortex dipole containing two vortex cores. If there is a large difference in their initial strengths, there is a large difference in the strengths of the resulted vortex cores. As a result, the weaker vortex rotates around the stronger one continuously, which is the key source of the aerodynamic sound. This DNS result provides a new concept: the interactions among turbulent structures with large differences in their scales play a key role in the generation of turbulent noise.

We have investigated high order finite difference weighted essentially non-oscillatory (WENO) schemes, finite volume WENO schemes and discontinuous Galerkin finite element methods, for solving partial differential equations with discontinuous or rapidly changing solutions. Algorithm development, analysis, implementation and applications have been carried out. Research has been performed in all areas listed in the original proposal, and progress and results consistent with the original objectives have been obtained. There are 53 refereed journal publications (42 appeared, 11 accepted and to appear) resulting from this project. These achievements have strengthened our objective to obtain powerful and reliable high order numerical algorithms and use them to solve convection dominated problems, especially those of army interest.

In this paper, we study the strong stability preserving (SSP) property of a class of deferred correction time discretization methods, for solving the method-of-lines schemes approximating hyperbolic partial differential equations.

In this paper we develop a high order explicit finite difference weighted essentially non-oscillatory (WENO) scheme for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. The main technical complication is the existence of global terms in the coefficient and boundary condition for this model. We carefully design approximations to these global terms and boundary conditions to ensure high order accuracy. Comparing with the first order monotone and second order total variation bounded schemes for the same model, the high WENO scheme is more efficient and can produce accurate results with far fewer grid points. Numerical examples including one in computational biology for the evolution of the population of Gambussia affinis, are presented to illustrate the good performance of the high order WENO scheme.

In this paper, we further explore and apply our recent anti-diffusive flux corrected high order finite difference WENO schemes for conservation laws to compute the Saint-Venant system of shallow water equations with pollutant propagation, which is described by a transport equation. The motivation is that the high order anti-diffusive high order WENO scheme to the Saint-Venant system of shallow water equations with transport of pollutant achieves high resolution of the location and concentration of the pollutant.

In this paper, we explicitly construct the entropy solutions for the Lighthill-Whitham-Richards (LWR) traffic flow model with a flow-density relationship which is piecewise quadratic, continuous, concave, but not differentiable at the junction points where two quadratic polynomials meet, and with piecewise linear initial condition and piecewise constant boundary conditions. As observed traffic flow data can be well fitted with such continuous piecewise quadratic functions, the explicitly constructed solutions provide a fast and accurate solution tool which may be used for predicting traffic or as a diagnosing tool to test the performance of numerical schemes. We implement these explicit entropy solutions for three representative traffic flow cases and also compare them with numerical solutions obtained by a high order weighted essentially non-oscillatory (WENO) scheme.

This project is about the algorithm development, analysis, implementation and application aspects of high order finite difference weighted essentially non-oscillatory (WENO) schemes, finite volume WENO schemes, discontinuous Galerkin finite element methods and spectral methods for solving convection dominated problems requiring long time integration and small dissipation/dispersion with discontinuous or high gradient solutions. Algorithm development and analysis, investigation about efficient implementation including parallel implementations, and applications in computational fluid dynamics, computational semiconductor device simulation and other areas, are performed. The achievement strengthens our objective to obtain powerful and reliable high order numerical algorithms and use them to solve convection dominated problems, especially those of army interest.

This project is about the design, analysis and application of high order accurate and nonlinearly stable finite difference (including finite volume) WENO schemes, finite element discontinuous Galerkin methods, and spectral algorithms for computing solutions of partial differential equations which are either discontinuous or with sharp gradients. Algorithm development, theoretical study about stability and convergence of the algorithms, investigation about efficient implementation including parallel implementations, and applications in compressible and incompressible gas dynamics and in semiconductor device simulations. are performed. The achievement strengthens our objective to obtain powerful and reliable high order numerical algorithms and use them to solve problems containing discontinuous solutions, especially those of Army interest.

The overarching goal of this research was to construct stable, robust and efficient high order accurate computational methods for long time integration of nonlinear partial differential equations. High order accuracy methods (Spectral, Finite Difference and Finite Elements) for the numerical simulations of flows with discontinuities, in complex geometries were developed. In particular applications in supersonic combustion were emphasized. Specific research subjects included: Robust high order compact difference schemes, ENO and WENO schemes, discontinuous Galerkin methods, the resolution of the Gibbs phenomenon, parallel computing and high order accurate boundary conditions. In order to overcome the difficulties stemming from complicated geometries, we have developed multidomain techniques as well as spectral methods on arbitrary grids. Several multidimensional codes for supersonic reactive flows had been constructed as well as a library of spectral codes (Pseudopack).

This project is about the design, analysis and application of high order accurate and nonlinearly stable finite difference (including finite volume), finite element and spectral algorithms for computing solutions of partial differential equations which are either discontinuous or with sharp gradients. Algorithm development, theoretical study about stability and convergence of the algorithms, investigation about efficient implementation including parallel implementations, and applications in compressible and incompressible gas dynamics and in semiconductor device simulations, are performed. The achievement strengthens our objective to obtain powerful and reliable high order numerical algorithms and use them to solve problems containing discontinuous solutions, especially those of army interest.

In this research we presented the development and application of spectral shock capturing techniques as well as ENO finite difference and finite element methods for realistic problems. For genuinely time dependent problems, special care had to be taken in order to preserve accuracy. Small errors that do not show up in steady state calculations can be amplified and ruin the total accuracy. Time dependent boundary conditions may also pose problems. We addressed those issues in our research efforts. The recent advent of parallel computers poses a special challenge to the users of high-order methods. Issues in parallel computing for the simulations of incompressible and compressible flows has been treated.