9 research outputs found
Arc length based WENO scheme for Hamilton-Jacobi Equations
In this article, novel smoothness indicators are presented for calculating
the nonlinear weights of weighted essentially non-oscillatory scheme to
approximate the viscosity numerical solutions of Hamilton-Jacobi equations.
These novel smoothness indicators are constructed from the derivatives of
reconstructed polynomials over each sub-stencil. The constructed smoothness
indicators measure the arc-length of the reconstructed polynomials so that the
new nonlinear weights could get less absolute truncation error and gives a
high-resolution numerical solution. Extensive numerical tests are conducted and
presented to show the performance capability and the numerical accuracy of the
proposed scheme with the comparison to the classical WENO scheme.Comment: 14 pages, 9 figure
A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations
In this paper, we improve upon the discontinuous Galerkin (DG) method for
Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W.
Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for
directly solving the general HJ equations. The new method avoids the
reconstruction of the solution across elements by utilizing the Roe speed at
the cell interface. Besides, we propose an entropy fix by adding penalty terms
proportional to the jump of the normal derivative of the numerical solution.
The particular form of the entropy fix was inspired by the Harten and Hyman's
entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983)
for Roe scheme for the conservation laws. The resulting scheme is compact,
simple to implement even on unstructured meshes, and is demonstrated to work
for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension
and two dimensions are provided to validate the performance of the method
A quasi-conservative discontinuous Galerkin method for multi-component flows using the non-oscillatory kinetic flux
In this paper, a high order quasi-conservative discontinuous Galerkin (DG)
method using the non-oscillatory kinetic flux is proposed for the 5-equation
model of compressible multi-component flows with Mie-Gr\"uneisen equation of
state. The method mainly consists of three steps: firstly, the DG method with
the non-oscillatory kinetic flux is used to solve the conservative equations of
the model; secondly, inspired by Abgrall's idea, we derive a DG scheme for the
volume fraction equation which can avoid the unphysical oscillations near the
material interfaces; finally, a multi-resolution WENO limiter and a
maximum-principle-satisfying limiter are employed to ensure oscillation-free
near the discontinuities, and preserve the physical bounds for the volume
fraction, respectively. Numerical tests show that the method can achieve high
order for smooth solutions and keep non-oscillatory at discontinuities.
Moreover, the velocity and pressure are oscillation-free at the interface and
the volume fraction can stay in the interval [0,1].Comment: 41 pages, 70 figure
Adaptive sparse grid discontinuous Galerkin method: review and software implementation
This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG)
method for computing high dimensional partial differential equations (PDEs) and
its software implementation. The C\texttt{++} software package called AdaM-DG,
implementing the aSG-DG method, is available on Github at
\url{https://github.com/JuntaoHuang/adaptive-multiresolution-DG}. The package
is capable of treating a large class of high dimensional linear and nonlinear
PDEs. We review the essential components of the algorithm and the functionality
of the software, including the multiwavelets used, assembling of bilinear
operators, fast matrix-vector product for data with hierarchical structures. We
further demonstrate the performance of the package by reporting numerical error
and CPU cost for several benchmark test, including linear transport equations,
wave equations and Hamilton-Jacobi equations
Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations
The dissertation focuses on numerically approximating viscosity solutions to second order fully nonlinear partial differential equations (PDEs). The primary goals of the dissertation are to develop, analyze, and implement a finite difference (FD) framework, a local discontinuous Galerkin (LDG) framework, and an interior penalty discontinuous Galerkin (IPDG) framework for directly approximating viscosity solutions of fully nonlinear second order elliptic PDE problems with Dirichlet boundary conditions. The developed frameworks are also extended to fully nonlinear second order parabolic PDEs. All of the proposed direct methods are tested using Monge-Ampere problems and Hamilton-Jacobi-Bellman (HJB) problems. Due to the significance of HJB problems in relation to stochastic optimal control, an indirect methodology for approximating HJB problems that takes advantage of the inherent structure of HJB equations is also developed.
First, a FD framework is developed that guarantees convergence to viscosity solutions when certain properties concerning admissibility, stability, consistency, and monotonicity are satisfied. The key concepts introduced are numerical operators, numerical moments, and generalized monotonicity. One class of FD methods that fulfills the framework provides a direct realization of the vanishing moment method for approximating second order fully nonlinear PDEs. Next, the emphasis is on extending the FD framework using DG methodologies. In particular, some nonstandard LDG and IPDG methods that utilize key concepts from the FD framework are formulated. Benefits of the DG methodologies over the FD methodology include the ability to handle more complicated domains, more freedom in the design of meshes, higher potential for adaptivity, and the ability to use high order elements as a means for increased accuracy. Last, a class of indirect methods for approximating HJB equations using the vanishing moment method paired with a splitting formulation of the HJB problem is developed and tested numerically. The proposed methodology is well-suited for both continuous and discontinuous Galerkin methods, and it complements the direct methods developed in the dissertation