27 research outputs found
A hybrid Hermite WENO scheme for hyperbolic conservation laws
In this paper, we propose a hybrid finite volume Hermite weighted essentially
non-oscillatory (HWENO) scheme for solving one and two dimensional hyperbolic
conservation laws. The zeroth-order and the first-order moments are used in the
spatial reconstruction, with total variation diminishing Runge-Kutta time
discretization. The main idea of the hybrid HWENO scheme is that we first use a
shock-detection technique to identify the troubled cell, then, if the cell is
identified as a troubled cell, we would modify the first order moment in the
troubled cell and employ HWENO reconstruction in spatial discretization;
otherwise, we directly use high order linear reconstruction. Unlike other HWENO
schemes, we borrow the thought of limiter for discontinuous Galerkin (DG)
method to control the spurious oscillations, after this procedure, the scheme
would avoid the oscillations by using HWENO reconstruction nearby
discontinuities and have higher efficiency for using linear approximation
straightforwardly in the smooth regions. In addition, the hybrid HWENO scheme
still keeps the compactness. A collection of benchmark numerical tests for one
and two dimensional cases are performed to demonstrate the numerical accuracy,
high resolution and robustness of the proposed scheme.Comment: 38 page
Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes
This is the continuation of the paper "central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction" by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to nite volume schemes on non-staggered grids. This takes a new nite volume approach for approximating non-smooth solutions. A critical step for high order nite volume schemes is to reconstruct a nonoscillatory
high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed
polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th order schemes without
characteristic decomposition.The research of Y. Liu was supported in part by NSF grant DMS-0511815. The research of C.-W. Shu was supported in part by the Chinese Academy of Sciences while this author was visiting the University of Science
and Technology of China (grant 2004-1-8) and the Institute of Computational Mathematics and Scienti c/Engineering Computing. Additional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS-0510345. The research of E. Tadmor was supported in part by NSF grant 04-07704 and ONR grant N00014-91-J-1076. The research of M. Zhang was supported in part by the Chinese Academy of Sciences grant 2004-1-8
A gradient-augmented level set method with an optimally local, coherent advection scheme
The level set approach represents surfaces implicitly, and advects them by
evolving a level set function, which is numerically defined on an Eulerian
grid. Here we present an approach that augments the level set function values
by gradient information, and evolves both quantities in a fully coupled
fashion. This maintains the coherence between function values and derivatives,
while exploiting the extra information carried by the derivatives. The method
is of comparable quality to WENO schemes, but with optimally local stencils
(performing updates in time by using information from only a single adjacent
grid cell). In addition, structures smaller than the grid size can be located
and tracked, and the extra derivative information can be employed to obtain
simple and accurate approximations to the curvature. We analyze the accuracy
and the stability of the new scheme, and perform benchmark tests.Comment: 28 pages, 14 figure
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions
Multiresolution strategies for the numerical solution of optimal control problems
Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme.
The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a nonlinear programming (NLP) problem that is solved using standard NLP codes. The novelty of the proposed approach hinges on the automatic calculation of a suitable, nonuniform grid over which the NLP problem is solved, which tends to increase numerical efficiency and robustness. Control and/or state constraints are handled with ease, and without any additional computational complexity. The proposed algorithm is based on a simple and intuitive method to balance several conflicting objectives, such as accuracy of the solution, convergence, and speed of the computations. The benefits of the proposed algorithm over uniform grid implementations are demonstrated with the help of several nontrivial examples. Furthermore, two sequential multiresolution trajectory optimization algorithms for solving problems with moving targets and/or dynamically changing environments have been developed.Ph.D.Committee Chair: Tsiotras, Panagiotis; Committee Member: Calise, Anthony J.; Committee Member: Egerstedt, Magnus; Committee Member: Prasad, J. V. R.; Committee Member: Russell, Ryan P.; Committee Member: Zhou, Hao-Mi
High-order methods for diffuse-interface models in compressible multi-medium flows: a review
The diffuse interface models, part of the family of the front capturing methods, provide an efficient and robust framework for the simulation of multi-species flows. They allow the integration of additional physical phenomena of increasing complexity while ensuring discrete conservation of mass, momentum, and energy. The main drawback brought by the adoption of these models consists of the interface smearing, increasing with the simulation time, therefore, requiring a counteraction through the introduction of sharpening terms and a careful selection of the discretization level. In recent years, the diffuse interface models have been solved using several numerical frameworks including finite volume, discontinuous Galerkin, and hybrid lattice Boltzmann method, in conjunction with shock and contact wave capturing schemes. The present review aims to present the recent advancements of high-order accuracy schemes with the capability of solving discontinuities without the introduction of numerical instabilities and to put them in perspective for the solution of multi-species flows with the diffuse interface method.Engineering and Physical Sciences Research Council (EPSRC): 2497012.
Innovate UK: 263261.
Airbus U
High-resolution numerical schemes for compressible flows and\ud compressible two-phase flows
Several high-resolution numerical schemes based on the Constrained Interpolation Profile
Conservative Semi-Lagrangian (CIP-CSL), Essentially Non-Oscillatory (ENO),
Weighted ENO (WENO), Boundary Variation Diminishing (BVD), and Tangent of
Hyperbola for INterface Capturing (THINC) schemes have been proposed for compressible
flows and compressible two-phase flows.
In the first part of the thesis, three high-resolution CIP-CSL schemes are proposed.
(i) A fully conservative and less oscillatory multi-moment scheme (CIP-CSL3-ENO)
is proposed based on two CIP-CSL3 schemes and the ENO scheme. An ENO indicator
is designed to intentionally select non-smooth stencil but can efficiently minimise
numerical oscillations. (ii) Motivated by the observation that combining two different
types of reconstruction functions can effectively reduce numerical diffusion and
oscillations, a better-suited scheme CIP-CSL-ENO5 is proposed based on hybrid-type
CIP-CSL reconstruction functions and a newly designed ENO indicator. (iii) To further
reduce the numerical diffusion in vicinity of discontinuities, the BVD and THINC
schemes are implemented in the CIP-CSL framework. The resulting scheme accurately
capture both smooth and discontinuous solutions simultaneously by selecting an
appropriate reconstruction function.
In the second part of the thesis, the TWENO (Target WENO) scheme is proposed to
improve the accuracy of the fifth-order WENO scheme. Unlike conventional WENO
schemes, the TWENO scheme is designed to restore the highest possible order interAbstract
iv
polation when three sub-stencils or two adjacent sub-stencils are smooth. To further
minimise the numerical diffusion across discontinuities, the TWENO scheme is implemented
with the THINC scheme and the Total Boundary Variation Diminishing
(TBVD) algorithm. The resulting scheme TBVD-TWENO-THINC is also applied to
solve the five-equation model for compressible two-phase flows.
Verified through a wide range of benchmark tests, the proposed numerical schemes are
able to obtain accurate and high-resolution numerical solutions for compressible flows
and compressible two-phase flows
The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis
The vanishing moment method was introduced by the authors in [37] as a
reliable methodology for computing viscosity solutions of fully nonlinear
second order partial differential equations (PDEs), in particular, using
Galerkin-type numerical methods such as finite element methods, spectral
methods, and discontinuous Galerkin methods, a task which has not been
practicable in the past. The crux of the vanishing moment method is the simple
idea of approximating a fully nonlinear second order PDE by a family
(parametrized by a small parameter \vepsi) of quasilinear higher order (in
particular, fourth order) PDEs. The primary objectives of this book are to
present a detailed convergent analysis for the method in the radial symmetric
case and to carry out a comprehensive finite element numerical analysis for the
vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract
methodological and convergence analysis frameworks of conforming finite element
methods and mixed finite element methods are first developed for fully
nonlinear second order PDEs in general settings. The abstract frameworks are
then applied to three prototypical nonlinear equations, namely, the
Monge-Amp\`ere equation, the equation of prescribed Gauss curvature, and the
infinity-Laplacian equation. Numerical experiments are also presented for each
problem to validate the theoretical error estimate results and to gauge the
efficiency of the proposed numerical methods and the vanishing moment
methodology.Comment: 141 pages, 16 figure