The development of limiting techniques starts from high resolution capturing schemes for solving nonlinear conservation laws whose weak solutions contain discontinuities. These schemes do not trace discontinuities in a weak solution individually and automatically smear them into transition layers within a few mesh cells. They can achieve high order of accuracy if the solution is smooth and there is a nonlinear limiting mechanism to prevent spurious oscillations in the vicinities of discontinuities. The limiting techniques have since been developed for many other methods and applications, e.g., the Runge-Kutta discontinuous Galerkin methods with limiting, the moment limiter etc. Hierarchical reconstruction decomposes the job of limiting a high degree polynomial defined in a cell into a series of smaller jobs, each of which only involves the non-oscillatory reconstruction of a linear polynomial from cell averages. Therefore it only uses information from adjacent cells and can be naturally formulated on unstructured meshes in multi dimensions. It does not use local characteristic decomposition and thus is less dependent on the underlying equation to be solved. The principle investigator proposes several new improvements related to the hierarchical reconstruction in higher orders. The analytical study of the role of the remainder term in it could provide deeper understanding of the limiting mechanism. In particular, a compact, multi-step method is proposed to reconstruct a piecewise polynomial function of high degree from cell averages and sparsely located polynomial approximations. This property is novel. Its development and theoretical understanding is a new area to be explored.
More and more complex problems from science, engineering, business and daily life are handled by computers. However, only a finite amount of information can be stored and all numbers are truncated in a computer with a finite number of digits before and after being processed. Therefore a computer simulation is an approximation and is usually "noisy" as in the real world. In particular, non-smooth data tends to induce artifacts in computational solutions, making them less useful or completely useless. Non-smooth data is common in real applications. For example, the air pressure and density have jumps across a shockwave induced by a supersonic aircraft; the human body contains various jumps in density; in nanoscience, fuel cells, composite materials, material defect detection etc, non-smooth data originates from interfaces between different materials, irregular boundaries and cracks; in simulations in environmental science, ocean and atmosphere, non-smooth data comes from heterogeneous underground structures, irregular seafloor, seashore and ground surface, dynamic interfaces separating solid, liquid and gas etc. The project involves the development and analysis of a general method which eliminates as much computational artifacts as possible from the underlying solution without actually knowing it. The proposed limiting techniques are less problem dependent and can be useful in solving gas dynamics equations, magnetohydrodynamics equations and many other equations related to these applications. The new compact, multi-step reconstruction method could significantly reduce the memory cost of the discontinuous Galerkin methods enabling them to solve more complicated applications. It can also be formulated as a compact interpolation method and can be broadly used in computer graphics, image processing and many other scientific and engineering computations.
The PI and his collaborators have made several interesting findings and innovations with the NSF support. (1) A new constrained DG method which has about the same complexity as the popular RKDG method but improves its time step size 3 times or more. This work has been accepted by Journal of Computational Physics. Intellectual merit: The new method reformulates an RKDG method as the minimization of a quadratic energy functional, thus allowing adding extra penalty terms to the energy functional to improve the CFL numbers without hurting the order of accuracy. A new technique is introduced without using Lagrangian multipliers so that the complexity won't increase. This idea opens up a new area for improving a class of conventional computational methods. Broader impacts: The RKDG method has been widely used in applications such as in the computation of air flow, ocean flow, bio fluid etc. The new findings help improve the efficiency of the computation. The idea can also be applied to other variants of DG methods, which will be explored in the future. (2) A new limiting strategy for the BFECC method for solving Hamilton-Jacobi equations. This work has been reviewed by Mathematics of Computation. Intellectual merit: BFECC method was introduced by Dupont and the PI in 2003. It improves the order of accuracy of a low order method by essentially calling it 3 times, and has since been used in various applications such as fluid simulations for graphics and animations. The new limiting method developed for BFECC allows it to compute non smooth solutions with less artifacts, such as near fluid interfaces or from fluid-solid interaction. The idea is novel and conventional approaches don't seem to fit the method naturally. The PI, his Ph.D student and a collaborator also explore the computation of Hamilton-Jacobi equations on triangular meshes with BFECC with good results and ease of implementation. Broader impacts: Real application of Hamilton-Jacobi equations often involves complex geometry for which a triangular mesh is usually more suitable. The PI and his collaborators explore a convenient second order implementation of BFECC on triangular meshes for the equations which essentially calls a first order subroutine three times. The newly developed limiting technique allows the method to be applied to more applications with non smooth solutions, and makes further improvement in previous applications in fluid simulations for computer graphics and animations. The new technique is unconventional, and could motivate more developments in the area. (3) A constant subtraction technique for solving shallow water equation with central schemes on overlapping cells. This work has been accepted by Journal of Scientific Computing. Intellectual merit: The PI, his Ph.D student and a collaborator develop a very convenient new technique to solve the shallow water equation and balanced laws with central schemes on overlapping cells, making them well-balanced schemes. The technique makes adjustment in the equation rather than in the scheme, thus is quite general and can be used to turn other numerical schemes into well-balanced schemes. Also a modified limiting technique (based on Hierarchical Reconstruction, developed by the PI and his collaborators with NSF support for removing numerical artifacts in computing non smooth solution) is developed for removing numerical artifacts induced by discontinuities of solutions. Broader impacts: Shallow water equation models water surface dynamics in the ocean, lakes, rivers etc. The techniques developed in the work enable a very convenient adaptation of commonly used numerical methods for conservation laws to shallow water equations and other balanced laws with good accuracy when their solutions are close to stationary states. The attractive feature is that the technique is very easy to implement and is fairly general. The PI and his collaborators are extending the technique to other balanced laws from various applications.
