Xu-Dong Liu - US grants

DMS-9805546 Xu-Dong Liu This project is concerned with several new numerical methods for solving multidimensional systems of conservation laws, including a new fully conservative scheme for multi-phase fluid calculations. The first contribution of this project is the extension of Friedrich's positivity principle from multi-dimensional symmetric linear systems to systems of conservation laws, which has become one of the guidelines for designing numerical methods. A family of positive schemes is constructed. Positive schemes are very robust, simple and of low cost. Many numerical experiments have shown that positive schemes are among the best 2nd order accurate high resolution methods. This is also the first work of this type which contains theoretical results for scheme design in multi-dimensional hyperbolic systems. The second contribution of this project is the new Convex Essential-Non-Oscillatory (CENO) schemes for multi-dimensional hyperbolic systems. The scheme can be implemented in component-wise fashion. Therefore its apparent advantages are: (1) No complete set of eigenvectors is needed and hence weakly hyperbolic systems can be solved. (2) Component-wise limiting is twice as fast as field-by-field limiting in each space dimension, which makes Convex ENO one of the fastest existing schemes. (3) The component-wise version of the scheme is simple to program. In addition, (4) the Convex ENO scheme is very robust. The third contribution of this project is the introduction of a fully conservative method for multi-phase flow problems. This new idea enables us to avoid the spurious oscillations near material interfaces common to all other conservative schemes. This is done through the addition of a general equation of state. The new scheme works essentially for mixture of any fluids such as gamma-law gas, water and JWL (explosive material). The new idea works in any space dimension and is scheme-independent, which means it should apply to a typical users' existi ng code. Preliminary numerical experiments show that this scheme is very promising. This project is aimed at solving real world problems and is intended to have a significant impact on semiconductor device modeling, underwater and solid explosives modeling, computational fluid dynamics, magneto-hydrodynamics, and many other applications, which are all a part of high-performance computing. The principal goals are: (1) to design and improve numerical methods for more efficiency, simplicity, and robustness; (2) to improve computer simulation of multi-phase fluids. The methods reported in this project are a step towards this goal.

The main theme of the proposed project is the construction of high order accurate numerical schemes for solving multi-dimensional hyperbolic systems of conservation laws, and in particular the construction of numerical schemes for simulations of multi-phase fluid flows. This includes numerical methods for compressible flow, incompressible flow and heat transfer. Recently, the PI's introduced a boundary condition capturing method for variable coefficient Poisson equation in the presence of interfaces. The method is implemented using a standard finite difference discretization on a Cartesian grid making it simple to apply in several spatial dimensions. Furthermore, the resulting linear system is symmetric positive definite allowing for straightforward application of standard "black box" solvers, for example, multi-grid methods. Most importantly, this new method does not suffer from the numerical smearing. Using this method, the PI's extended the Ghost Fluid Method to treat two-phase incompressible flows, in particular those consisting of water and air. The numerical experiments show that the new numerical method performs quite well in both two and three spatial dimensions. Currently, they are working on extending this method to treat a wide range of problems, including for example combustion. Of particular interest is the extension of this method to include interface motion governed by the Cahn-Hilliard equation which models the non-zero thickness interface with a molecular force balance model.

This proposed research on computational fluid dynamics is focused on the design, implementation and testing of new methods for simulating fluids such as water and gas using the computer. In particular, this work addresses problems where more than one type of one phase of fluid exist, e.g. mixtures of water and air. Our interest lies in improving the current state of the art algorithms so that they are better able to treat the interface that separates two fluids such as oil and water. The results of this research should be of interest to both the military (e.g. many naval applications involve the study of water and air mixtures) and to private industry. A particularly interesting example involves the interaction of water and oil in an underground oil recovery process. The research covered in this proposal has implications for math and science education as well. Not only will the PI's be working with and training graduate students in applied mathematics and engineering, but their research in extending these techniques to other fields, such as computer graphics, can play a role attracting the next generation of young scientists. For example, figure 7 in "Foster and Fedkiw, Practical Animation of Liquids, SIGGRAPH 2001" shows the lovable character "Shrek", from the feature film of the same name, taking a bath in mud.

The Department of Mathematics at the University of California, Santa
Barbara will purchase a Beowulf Cluster consisting of 16 dual processor
node, one single processor controlling node, a network switch, backup tape
drive, and a rack. This hardware will be dedicated to the support of
research in the mathematical sciences. The equipment will be used for
several research projects, including in particular: Computations of
scaling of fluvial landscapes, computations of effectively nonlinear
quantum systems, dynamically adaptive and nonstiff boundary integral
methods, numerical schemes for simulations of multi-phase fluids and
vorticity deformation in 2D ideal incompressible fluid flow.

In a series of research works we have introduced and established the
positivity principle for schemes for solving hyperbolic systems of conservation
laws. The rationale of the positivity principle is stability, which is a very
important requirement for numerical schemes. The positivity principle is the
1st stability principle for schemes for solving multi-dimensional hyperbolic
systems. In this proposal we have shown that the central scheme studied by Kurganov and Tadmor is positive. By mixing upwind scheme and Lax-Wendroff scheme, we have made a positive scheme which costs only 30% of the original positive scheme. We have developed a scheme called Convex Essentially Non-Oscillatory (ENO) scheme. The Convex ENO scheme is a high order accurate central scheme. We have developed a new multigrid method to solve hyperbolic systems of conservation laws. By doing multigrid, the cost of calculations is reduced significantly. In the proposal we also develop several schemes for solving elliptic problems with multi-fluids separated by the interfaces. Such problems arise from many real world applications. For example, incompressible multi-fluids Navier-Stokes equation. A new uniform 2nd order accurate scheme on non-body-fitting grids is developed for that. We have proposed a uniform 2nd order accurate level-set method using finite element method for solving elliptic problems with mixing boundary conditions. Such problems emerge from in simulating epitaxial thin film growth using the island dynamics model. We have used some of those methods to do Direct Numerical Simulation on multi-phase turbulent flows. We have developed a geometric multigrid method for such elliptic problems based on the Ghost Fluid Method, and plan to do more with the other methods. The PI and his collaborators are pursuing further development of positive schemes. In a series of research works they have introduced and established the positivity principle for schemes for solving hyperbolic systems of conservation laws. The rationale of the positivity principle is stability. 1) They prove that the central scheme developed by Kurganov and Tadmor is positive scheme. 2) They continue to develop a new positive scheme, which is a mixture of upwind and Lax-Wendroff schemes. Hence two-stage Runge-Kutta is not required and for two-dimensions the computation cost could be cut by as much as 70%.
3) They continue to work on a new scheme called weighted component-wise positive
scheme. It is a mixture of Weighted ENO schemes and 2nd order component-wise version of Convex ENO scheme or high-resolution central scheme. They use a convex combination of all candidates to do reconstruction, but use a new measurement called accurateness instead of smoothness to assign proper weights. The convex combination achieves almost optimal (one order lower than the optimal) order accuracy. This scheme can be also extended to solve Hamilton-Jacobi equations in multi-dimensions. 4) They are going to introduce a multigrid method for solving multi-dimensional hyperbolic systems of conservation laws. The novelty is to calculate the fluxes on coarse grid, then interpolates the differences of the fluxes or the fluxes to the finest grid. Such multigrid method is not only faster than a base scheme in each iteration, but also allows larger time step than that of the base scheme. Hence the multigrid method requires much less CPU time to advance solutions to the same stopping time compared to the base scheme. In other words, for the same CPU time, the multigrid method advances solutions much further in time. This is particularly useful for computing stationary solutions. In the recent years, the PI and his collaborators have been pursuing further development of Ghost Fluid Method (GFM) for multi-phase fluids. 1) They propose a geometric multigrid method to solving linear systems arising from irregular boundary problems involving multiple interfaces in 2D and 3D. In this method, they adopt a matrix-free approach i.e. they do not form the fine grid matrix explicitly and they never form nor store the coarse grid matrices. The main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function. 2) They propose a 2nd order accurate level-set method using finite element method for solving elliptic equations with Robin interface conditions. They first study a weak formulation of it, and then prove that
there exists a unique weak solution. At last, a finite element method on non-body-fitting uniform or arbitrary triangulations is used to solve such weak formulation. The novelty of this work is the incorporation of finite element methods and non-body-fitting triangulations. 3) They develop a new 2nd order accurate numerical method on non-body-fitting grids for solving the elliptic equations with interfaces. Instead of smooth, the boundary and the subdomains'
boundaries and hence the interfaces, are only required to be Lipschitz continuous as submanifold. A weak formulation is developed and the numerical method is derived by discretizing the weak formulation by piece-wise linear functions. The method is 2nd order accurate in maximum norm if the interface is smooth or its discontinuities are proper handled, and convergent in maximum norm otherwise. 4) They use the boundary condition capturing method
to do Direct Numerical Simulations on multi-phase turbulent
flows. This is the first successful DNS of such problems.
Because turbulence happens through a large range of scales, and hence very efficient methods are needed to capture all meaningful scales.

The proposal focus on the real world applications. For example, hyperbolic
systems of conservation laws, incompressible Navier-Stokes equations with interfaces, epitaxial thin film growth using the island dynamics model, Direct Numerical Simulation on multi-phase turbulent flows. The proposed numerical methods possess high order accuracy and high resolutions, hence they are very efficient. Two multigrid methods are proposed to couple with those methods to further speeded up numerical simulations. The proposal should have broad impact, since the methods created can be easily adopted to many other application areas in the environmental, geophysical, biological, material science, and engineering sciences.

This project focuses on the development of a computational toolbox for investigation of multiscale
surface processes that are central to nanotechnology as well as other current technologies. Two physical
systems will be studied that span from nano-scale phenomena to large-scale deterministic transport
phenomena. The algorithms and software, developed to simulate and extract information from multiscale
systems, are generic over a broad class of problems, and will contribute well beyond the applications used
in their development.
The physical systems include electrodeposition of metallic nanoclusters with additives to achieve
specific shapes, and environmental degradation through interaction of pits, crevices and cracks. The
physical systems, chosen for their computational structure, are characteristic of a large class of systems
where controlled shape evolution is exploited to produce desired structures. Key issues are to understand
how small-scale surface interactions guide spontaneous self-organization, how to extract insight from
noisy data and uncertain fundamental understanding, and how to insure quality control at multiple scales
in manufacturing.
Computational tools will be developed for simulation and sensitivity analysis in multi-phenomena
multiscale systems that require methods for coupling of stochastic and deterministic models. Challenges
for deterministic simulation include the effective use of parallel computers, and dealing with moving
boundaries, ill-conditioning and stiffness. We will explore classes of preconditioners for the iterative
methods that solve large linear systems of equations at each time step, in particular a newly-developed
multigrid method that is well-suited to moving boundary problems. Challenges for stochastic simulation
include stiffness, which has only recently been recognized as a barrier to efficiency for stochastic
simulation.
Sensitivity analysis is an important part of this effort. For the deterministic computations, we will
make use of recently developed methods that are adaptive in space and time. We will develop new
sensitivity analysis methods and software for stochastic systems, and couple them to deterministic
sensitivity analysis for the physical systems of interest. We will facilitate the use of our toolkit by
extension to larger-scale software systems of a recently-developed environment for the rapid creation of
GUIs for scientific and numerical software.
This project addresses the National Priority Area of Advanced Science and Engineering (ASE), and
the Technical Focus Areas of Innovation in computational modeling or simulation in research or
education (sim) (primary), and of Innovative approaches to the integration of data, models,
communications, analysis and/or control systems, including dynamic, data-driven applications for use in
prediction, risk-assessment and decision-making (dmc) (secondary).
Broader Impacts
The proposed project will impact the National Priority Area of ASE through the development of
algorithms and software to enhance the use of high performance computers in the investigation of
multiscale surface processes. The availability of such a toolbox will accelerate fundamental scientific
research and engineering design in an area with the potential for large economic impact. Software
developed as a result of this project will be widely distributed in the scientific and engineering, computer
science and mathematical sciences communities.
The educational activities feature a multidisciplinary, cross-institutional approach to graduate
education. Students will work in multidisciplinary teams, with joint thesis advisors from a primary and a
secondary discipline. This approach has recently been undertaken at UCSB with some success; we plan
to institutionalize this approach to graduate education in Computational Science and Engineering (CSE)
at UCSB, and to export the model to UIUC. The model also includes industrial internships, career
development workshops, and mentoring of undergraduates. Both UIUC and UCSB have been pioneers
in developing graduate programs in CSE and have programs with a similar structure which will facilitate
the sharing of educational ideas and innovations across the institutions.