In this paper, we consider Maxwell’s equations in linear dispersive media described by a
single-pole Lorentz model for electronic polarization. We study two classes of commonly
used spatial discretizations: finite difference methods (FD) with arbitrary even order
accuracy in space and high spatial order discontinuous Galerkin (DG) finite element
methods. Both types of spatial discretizations are coupled with second order semi-implicit
leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion
analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification
of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of
dispersion error can be made taking into account the model parameters, and mesh sizes
in the design of the two types of schemes. This work is a continuation of our previous
research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7].
The results for the numerical dispersion analysis of the reduced linear model, considered
in the present paper, can guide us in the optimal choice of discretization parameters
for the more complicated and nonlinear models. The numerical dispersion analysis of
the fully discrete FD and DG schemes, for the dispersive Maxwell model considered
in this paper, clearly indicate the dependence of the numerical dispersion errors on
spatial and temporal discretizations, their order of accuracy, mesh discretization parameters
and model parameters. The results obtained here cannot be arrived at by considering
discretizations of Maxwell’s equations in free space. In particular, our results contrast the
advantages and disadvantages of using high order FD or DG schemes and leap-frog or
trapezoidal time integrators over different frequency ranges using a variety of measures
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A mass- and energy-conserved DG method for the Schrödinger-Poisson equation
In this paper, we construct, analyze, and numerically validate conservative discontinuous Galerkin (DG) schemes for approximating the Schr\"{o}dinger-Poisson equation. The proposed
schemes all satisfy both mass and energy conservation. For the semi-discrete DG scheme optimal $L^2$ error estimates are obtained. Efficient iterative solvers are also constructed to solve the second
order implicit time discretization. A number of numerical tests are presented to demonstrate the method’s accuracy and robustness, confirming that both mass and energy are well preserved over long time simulations.
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- Award ID(s):
- 1812666
- NSF-PAR ID:
- 10282820
- Date Published:
- Journal Name:
- Numerical Algorithms
- ISSN:
- 1017-1398
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation. Experience has shown that if nothing special is done to account for the curl-preserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DG-like methods. The methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, highly crucial to the design of a DG method, is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work. The stability analysis confirms that with the increasing order of accuracy, our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation. We also show that P
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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.
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In this article, we introduce an invariant‐region‐preserving (IRP) limiter for the
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Sub-cloud rain evaporation in the trade wind region significantly influences the boundary layer mass and energy budgets. Parameterizing it is, however, difficult due to the sparsity of well-resolved rain observations and the challenges of sampling short-lived marine cumulus clouds. In this study, sub-cloud rain evaporation is analyzed using a steady-state, one-dimensional model that simulates changes in drop sizes, relative humidity, and rain isotopic composition. The model is initialized with relative humidity, raindrop size distributions, and water vapor isotope ratios (e.g., δDv, δ18Ov) sampled by the NOAA P3 aircraft during the Atlantic Tradewind Ocean–Atmosphere Mesoscale Interaction Campaign (ATOMIC), which was part of the larger EUREC4A (ElUcidating the RolE of Clouds–Circulation Coupling in ClimAte) field program. The modeled surface precipitation isotope ratios closely match the observations from EUREC4A ground-based and ship-based platforms, lending credibility to our model. The model suggests that 63 % of the rain mass evaporates in the sub-cloud layer across 22 P3 cases. The vertical distribution of the evaporated rain flux is top heavy for a narrow (σ) raindrop size distribution (RSD) centered over a small geometric mean diameter (Dg) at the cloud base. A top-heavy profile has a higher rain-evaporated fraction (REF) and larger changes in the rain deuterium excess (d=δD-8×δ18O) between the cloud base and the surface than a bottom-heavy profile, which results from a wider RSD with larger Dg. The modeled REF and change in d are also more strongly influenced by cloud base Dg and σ rather than the concentration of raindrops. The model results are accurate as long as the variations in the relative humidity conditions are accounted for. Relative humidity alone, however, is a poor indicator of sub-cloud rain evaporation. Overall, our analysis indicates the intricate dependence of sub-cloud rain evaporation on both thermodynamic and microphysical processes in the trade wind region.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11075-021-01139-0
