skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise
In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.  more » « less
Award ID(s):
1753581 2010107
PAR ID:
10416403
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
ESAIM: Mathematical Modelling and Numerical Analysis
Volume:
57
Issue:
2
ISSN:
2822-7840
Page Range / eLocation ID:
841 to 864
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, ESAIM: Mathematical Modelling and Numerical Analysis)
    In this paper, we study the error estimates to sufficiently smooth solutions of the nonlinear scalar conservation laws for the semi-discrete central discontinuous Galerkin (DG) finite element methods on uniform Cartesian meshes. A general approach with an explicitly checkable condition is established for the proof of optimal L 2 error estimates of the semi-discrete CDG schemes, and this condition is checked to be valid in one and two dimensions for polynomials of degree up to k = 8. Numerical experiments are given to verify the theoretical results. 
    more » « less
  2. (, Journal of computational physics)
    In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, we use a local scaling limiter to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations over long time. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes. 
    more » « less
  3. ; (, Mathematical Biosciences and Engineering)
    This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by using the Galerkin method combined with a compactness argument. A semi-discrete finite-element method and a fully discrete scheme using the Euler time-stepping have been formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented, along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed. 
    more » « less
  4. ; (, International Journal for Numerical Methods in Engineering)
    Abstract This article presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian–Eulerian formulation to map the wave equation from a time‐dependent moving physical domain onto a fixed reference domain. For moving curved meshes, weighted mass matrices must be assembled and inverted at each time step when using explicit time‐stepping methods. We avoid this step by utilizing an easily invertible weight‐adjusted approximation. The resulting semi‐discrete weight‐adjusted DG scheme is provably energy stable up to a term that (for a fixed time interval) converges to zero with the same rate as the optimal error estimate. Numerical experiments using both polynomial and B‐spline bases verify the high order accuracy and energy stability of proposed methods. 
    more » « less
  5. ; ; (, Numerical Methods for Partial Differential Equations)
    Abstract High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email