Journal of Computational Physics: X最新文献

We propose a three dimensional Discontinuous Petrov-Galerkin Maxwell approach for modeling Raman gain in fiber laser amplifiers. In contrast with popular beam propagation models, we are interested in a truly full vectorial approach. We apply the ultraweak DPG formulation, which is known to carry desirable properties for high-frequency wave propagation problems, to the coupled Maxwell signal/pump system and use a nonlinear iterative scheme to account for the Raman gain. This paper also introduces a novel and practical full-vectorial formulation of the electric polarization term for Raman gain that emphasizes the fact that the computer modeler is only given a measured bulk Raman gain coefficient. Our results provide promising qualitative corroboration of the model and methodology used.

Richardson's extrapolation process is a well known method to improve the order of several approximation processes. Here we observe that for numerical differentiation, Richardson's process can be applied not only to improve the order of a numerical differentiation formula but also to find in fact the original formula.

We present a multiscale continuous Galerkin (MSCG) method for the fast and accurate stochastic simulation and optimization of time-harmonic wave propagation through photonic crystals. The MSCG method exploits repeated patterns in the geometry to drastically decrease computational cost and incorporates the following ingredients: (1) a reference domain formulation that allows us to treat geometric variability resulting from manufacturing uncertainties; (2) a reduced basis approximation to solve the parametrized local subproblems; (3) a gradient computation of the objective function; and (4) a model and variance reduction technique that enables the accelerated computation of statistical outputs by exploiting the statistical correlation between the MSCG solution and the reduced basis approximation. The proposed method is thus well suited for both deterministic and stochastic simulations, as well as robust design of photonic crystals. We provide convergence and cost analysis of the MSCG method, as well as a simulation results for a waveguide T-splitter and a Z-bend to illustrate its advantages for stochastic simulation and robust design.

Conservation laws in the form of elliptic and parabolic partial differential equations (PDEs) are fundamental to the modeling of many problems such as heat transfer and flow in porous media. Many of such PDEs are stochastic due to the presence of uncertainty in the conductivity field. Based on the relation between stochastic diffusion processes and PDEs, Monte Carlo (MC) methods are available to solve these PDEs. These methods are especially relevant for cases where we are interested in the solution in a small subset of the domain. The existing MC methods based on the stochastic formulation require restrictively small time steps for high-variance conductivity fields. Moreover, in many applications the conductivity is piecewise constant and the existing methods are not readily applicable in these cases. Here we provide an algorithm to solve one-dimensional elliptic problems that bypasses these two limitations. The methodology is demonstrated using problems governed by deterministic and stochastic PDEs. It is shown that the method provides an efficient alternative to compute the statistical moments of the solution to a stochastic PDE at any point in the domain. A variance reduction scheme is proposed for applying the method for efficient mean calculations.

Similar to the obstacle or medium scattering problems, an important property of the phaseless far field patterns for source scattering problems is the translation invariance. Thus it is impossible to reconstruct the location of the underlying sources. Furthermore, the phaseless far field pattern is also invariant if the source is multiplied by any complex number with modulus one. Therefore, the source can not be uniquely determined, even the multi-frequency phaseless far field patterns are considered. By adding a reference point source into the model, we propose a simple and stable phase retrieval method and establish several uniqueness results with phaseless far field data. We proceed to introduce a novel direct sampling method for shape and location reconstruction of the source by using broadband sparse phaseless data directly. We also propose a combination method with the novel phase retrieval algorithm and the classical direct sampling methods with phased data. Numerical examples in two dimensions are also presented to demonstrate their feasibility and effectiveness.

The conditional hyperbolic quadrature method of moments (CHyQMOM) was introduced by Fox et al. [19] to reconstruct 1- and 2-D velocity distribution functions (VDF) from a finite set of integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a hyperbolic closure for the spatial flux term in the corresponding moment equations derived from a kinetic equation for the 3-D VDF. Here, CHyQMOM is extended for 3-D velocity phase space using the modified conditional quadrature method of moments with 16 (or 23) trivariate velocity moments up to fourth order. In order to verify the numerical implementation, it is applied to simulate several canonical particle-laden flows including crossing jets, cluster-induced turbulence (CIT), and vertical channel flow. The numerical results are compared with those from Euler–Lagrange simulations and two other quadrature-based moment methods, namely, anisotropic Gaussian (AG) and 8-node tensor-product (TP) quadrature. The relative advantages and disadvantages of each method are discussed. The crossing-jet problem highlights that CHyQMOM handles particle crossing more accurately than AG. For CIT, the results from all methods are similar, but the computational cost of TP is significantly larger than AG and CHyQMOM, both of which have nearly the same cost.

Macro-scale models of coupled free flow and flow through a permeable medium often lack the capabilities to account for process-relevant complexities on the pore scale. Direct numerical simulation of such systems, on the other hand, inherently includes these micro-scale features but is only feasible for problems of very limited spatial and temporal extent. A new class of hybrid models aims to combine the individual strengths, i.e., computational efficiency and local accuracy on the micro scale, of models of different dimensionality.

We propose, to our knowledge for the first time, a fully coupled model concept that involves a (Navier-) Stokes model for the free flow and a pore-network model for the porous domain. As a first step, we consider isothermal single-phase flow with and without component transport, but the model is open for extension for more complex physics. Appropriate coupling conditions guarantee the continuity of mass and momentum fluxes across the interface between the two domains. The coupled model is implemented in DuMu

The model is able to handle both structured and unstructured, randomly-generated networks. For the structured porous domains, the simulation results of the coupled model were compared to numerical reference solutions where excellent agreement was found, both for Reynolds numbers below one and around 400 in the free-flow channel. When applied to a geometrically complex unstructured network and considering compositional flow, clear paths of preferential flow could be identified which also locally affect the adjacent region of free flow at the respective interface. The ability to account for such pore-scale characteristics makes the model an interesting option, e.g., for simulating coupled flow problems that feature non-Fickian transport behavior or multi-phase flow, which will be investigated in future work.

While analytic calculations may give access to complex-valued electromagnetic field data which allow trivial access to envelope and phase information, the majority of numeric codes uses a real-valued representation. This typically increases the performance and reduces the memory footprint, albeit at a price: In the real-valued case it is much more difficult to extract envelope and phase information, even more so if counter propagating waves are spatially superposed. A novel method for the analysis of real-valued electromagnetic field data is presented in this paper. We show that, by combining the real-valued electric and magnetic field at a single point in time, we can directly reconstruct the full information of the electromagnetic fields in the form of complex-valued spectral coefficients ($\overrightarrow{k}$-space) at a low computational cost of only three Fourier transforms. The method allows for counter propagating plane waves to be accurately distinguished as well as their complex spectral coefficients, i.e. spectral amplitudes and spectral phase to be calculated. From these amplitudes, the complex-valued electromagnetic fields and also the complex-valued vector potential can be calculated from which information about spatiotemporal phase and amplitude is readily available. Additionally, the complex fields allow for efficient vacuum propagation allowing to calculate far field data or boundary input data from near field data. An implementation of the new method is available as part of PostPic¹, a data analysis toolkit written in the Python programming language.

An original strategy to address hydrodynamic flow was recently proposed through a high-order weakly-compressible Cartesian grid approach [1]. The method, named Weakly-Compressible Cartesian hydrodynamics (WCCH), is based on a fully-explicit temporal scheme for solving the Navier-Stokes equations while implicit incompressible schemes are usually preferred in the literature to address such flows. The present study aims to position and compare the WCCH method with a standard incompressible formulation. To this end, an incompressible scheme has been implemented in the same numerical framework. As far as possible, the algorithm used in the incompressible approach has been designed to be the same as (or close to) the one used in the weakly-compressible approach. In particular, high-order schemes for spatial and time discretization are employed. Pros and cons for each formulation are discussed in conjunction with a series of test cases on extensive criteria including implementation convenience, easy use of mesh refinement, convergence order and accuracy, numerical diffusion, parallel CPU scaling for high performance computing, etc. These comparisons demonstrate the relevance of the incompressible approach, at least for the selected test cases.

We study the Back and Forth Error Compensation and Correction (BFECC) method for linear hyperbolic systems and in particular for the Maxwell's equations. BFECC has been applied to schemes for scalar advection equations to improve their stability and order of accuracy. Similar results have been established in this paper for linear hyperbolic systems with constant coefficients. We apply BFECC to the central difference scheme, Lax-Friedrichs scheme and a combination of them for the Maxwell's equations and obtain second order accurate schemes with large CFL numbers (greater than 1 in one or two dimensions). The method is further applied to schemes on non-orthogonal unstructured grids. The new BFECC schemes for the Maxwell's equations operate on a single non-staggered grid and are simple to implement on unstructured grids. Numerical examples are given to demonstrate the effectiveness of the new schemes.