In this paper, we consider electromagnetic (EM) wave propagation in nonlinear optical media in one spatial dimension. We model the EM wave propagation by the time dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. The ODEs also include the single resonance linear Lorentz dispersion. For such model, we will design and analyze fully discrete finite difference time domain (FDTD) methods that have arbitrary (even) order in space and second order in time. It is challenging to achieve provable stability for fully discrete methods, and this depends on the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput Phys 350:420–452, 2017), we proposed novel modifications of secondorder leapfrog and trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize the nonlinear terms in this Maxwell model. Here, we continue this work by developing similar time discretizations within the framework of FDTD methods. More specifically, we design fully discrete modified leapfrog FDTD methods which are proved to be stable under appropriate CFLmore »
Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta timemarching for linear conservation laws
In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVDRK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasioptimal error estimates in space and optimal convergence rates in time. For the Eulerforward scheme with piecewise constant elements, the second order TVDRK method with piecewise linear elements and the third order TVDRK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Eulerforward scheme needs τ ≤ ρh 2 and the second order TVDRK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ more »
 Award ID(s):
 1719410
 Publication Date:
 NSFPAR ID:
 10093280
 Journal Name:
 ESAIM: Mathematical Modelling and Numerical Analysis
 Volume:
 53
 Issue:
 1
 Page Range or eLocationID:
 105 to 144
 ISSN:
 0764583X
 Sponsoring Org:
 National Science Foundation
More Like this


In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one and multidimensional cases with piecewise P k elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k ≥ 0. In particular, we adopt the techniques in Liu et al . ( SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P 1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.

The thermal radiative transfer (TRT) equations form an integrodifferential system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions TRT are challenging for several reasons, the first of which is that TRT is defined on a highdimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P$_N$ (spherical harmonics) or the S$_N$ (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H$^T_N$) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P$_N$ and S$_N$, and indeed reduces to each of these approximations in various limits: H$^1_N$ $\equiv$ P$_N$ and H$^T_0$ $\equiv$ S$_T$. We prove that H$^T_N$ results in a system of hyperbolic partial differential equations for all $T\ge 1$ and $N\ge 0$. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensivemore »

In this paper, we apply two fullydiscrete local discontinuous Galerkin (LDG) methods to the compressible wormhole propagation. We will prove the stability and error estimates of the schemes. Traditional LDG methods use the diffusion term to control of convection term to obtain the stability for some linear equations. However, the variables in wormhole propagation are coupled together and the whole system is highly nonlinear. Therefore, it is extremely difficult to obtain the stability for fullydiscrete LDG methods. To fix this gap, we introduce a new auxiliary variable including both the convection and diffusion terms. Moreover, we also construct a special time integration for the porosity, leading to physically relevant numerical approximations and controllable growth rate of the porosity. With a reasonable growth rate, it is possible to handle the time level mismatch in the firstorder fully discrete scheme and obtain the stability of the scheme. For the whole system, we will prove that under weak temporalspatial conditions, the optimal error estimates for the pressure, velocity, porosity and concentration under different norms can be obtained. Numerical experiments are also given to verify the theoretical results.

In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for timedependent 2D hyperbolic equations using P k elements on uniform Cartesian meshes, and prove that the error in the L 2 norm achieves optimal ( k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.