Equations In Heterogeneous Media Research Articles

An Adaptive and Quasi-periodic HDG Method for Maxwell’s Equations in Heterogeneous Media

With the aim to continue developing a hybridizable discontinuous Galerkin (HDG) method for problems arisen from photovoltaic cells modeling, in this manuscript we consider the time harmonic Maxwell’s equations in an inhomogeneous bounded bi-periodic domain with quasi-periodic conditions on part of the boundary. We propose an HDG scheme where quasi-periodic boundary conditions are imposed on the numerical trace space. Under regularity assumptions and a proper choice of the stabilization parameter, we prove that the approximations of the electric and magnetic fields converge, in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2$$\\end{document}-norm, to the exact solution with order hk+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1}$$\\end{document} and hk+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1/2}$$\\end{document}, resp., where h is the meshsize and k the polynomial degree of the discrete spaces. Although, numerical evidence suggests optimal order of convergence for both variables. An a posteriori error estimator for an energy norm is also proposed. We show that it is reliable and locally efficient under certain conditions. Numerical examples are provided to illustrate the performance of the quasi-periodic HDG method and the adaptive scheme based on the proposed error indicator.

Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media

We consider the time-harmonic Maxwell equations posed in R3. We prove a priori bounds on the solution for L∞ coefficients ϵ and μ satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of ϵ and μ. The class of coefficients covered includes (i) certain ϵ and μ for which well-posedness of the time-harmonic Maxwell equations had not previously been proved, and (ii) scattering by a penetrable C0 star-shaped obstacle where ϵ and μ are smaller inside the obstacle than outside. In this latter setting, the bounds are uniform across all such obstacles, and the first sharp frequency-explicit bounds for this problem at high-frequency.

An adaptive finite-difference method for seismic traveltime modeling based on 3D eikonal equation

3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing, source localization, reflection migration, seismic monitoring and tomographic imaging. In recent years, many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media. However, there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media. In this paper, we propose an adaptive finite-difference (AFD) method to numerically solve the 3D eikonal equation. The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes, and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle. Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media, our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts. Several numerical examples show the good performance of the AFD method, which will be beneficial to many scientific applications.

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Iterative Pure Source Transfer Domain Decomposition Methods for Helmholtz Equations in Heterogeneous Media

Iterative Pure Source Transfer Domain Decomposition Methods for Helmholtz Equations in Heterogeneous Media

Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than H2-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal.

A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media

AbstractIn this paper, we consider the development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well.

Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations

In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection diffusion reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and micro-scale equations, which do not contain memory terms. We note that dememorized equations differ from micro-scale equations and constitute a macroscopic model. Next, we consider both implicit and partially explicit methods. The latter is introduced for problems in multiscale media with high-contrast properties. Because of high-contrast, explicit methods are restrictive and require time steps that are very small (scales as the inverse of the contrast). We show that, by appropriately decomposing the space, we can treat only a few degrees of freedom implicitly and the remaining degrees of freedom explicitly. We present a stability analysis. Numerical results are presented that confirm our theoretical findings about partially explicit schemes applied to dememorized systems of equations.

On several numerical strategies to solve Richards’ equation in heterogeneous media with finite volumes

We propose several numerical approaches building on upstream mobility two-point flux approximation finite volumes to solve Richards’ equation in domains made of several rock-types. Our study encompasses four different schemes corresponding to different ways to approximate the nonlinear transmission condition systems arising at the interface between different rocks consisting in the continuity of the pressure and the fluxes, as well as different resolution strategies based on Newton’s method with variable switch. We benchmark the robustness and accuracy of the different methods on filling and drainage test cases with standard nonlinearities of Brooks-Corey and van Genuchten-Mualem type, as well as with challenging steep nonlinearities.

Elastic Correlative Least-Squares Reverse Time Migration Based on Wave Mode Decomposition

The conventional elastic least-squares reverse time migration (LSRTM) generally inverts the parameter perturbation of the model rather than the reflectivity of reflected P- and S-modes, which leads to difficulty in directly interpreting the physical properties of the subsurface media. However, an accurate velocity model that is needed by the separation of seismic records of conventional LSRTM is usually unavailable in real data, which limits its application. In this study, we introduce a new practical correlative LSRTM (CLSRTM) scheme based on wave mode decomposition without amplitude and phase distortion, which frees from separation of seismic records. In this study, we deduced the migration and the de-migration operators using the decoupled P- and S-wave equations in heterogeneous media, which needs no extra wavefield decomposition in simulated data. To accelerate the convergence and improve the efficiency of the inversion, we adopted an analytical step-length formula that can be incidentally computed during the necessary de-migration process and the L-BFGS algorithm. Two numerical examples demonstrate that the proposed method can compensate the energy of deep structures, and generate clear images with balanced amplitudes and enhanced resolution even for the fault structures beneath the salt dome.

Dual-pairing summation by parts finite difference methods for large scale elastic wave simulations in 3D complex geometries

High-order accurate summation-by-parts (SBP) finite difference (FD) methods constitute efficient numerical methods for simulating large-scale hyperbolic wave propagation problems. Traditional SBP FD operators that approximate first-order spatial derivatives with central-difference stencils often have spurious unresolved numerical wave-modes in their computed solutions. Recently derived high order accurate dual pair SBP operators based non-central (upwind) FD stencils have the potential to suppress these poisonous spurious wave-modes on marginally resolved computational grids. In this paper, we demonstrate that not all high order dual pair SBP FD operators are applicable. Numerical dispersion relation analysis shows that odd-order dual pair SBP FD operators also support spurious unresolved high-frequency wave-modes on marginally resolved meshes. Meanwhile, even-order dual pair SBP FD operators (of order 2,4,6) do not support spurious unresolved high frequency wave-modes and also have better numerical dispersion properties.For all the dual pair SBP FD operators we discretise the three space dimensional (3D) elastic wave equation on boundary-conforming curvilinear meshes. Using the energy method we prove that the semi-discrete approximation is stable and energy-conserving. We derive a priori error estimate and prove the convergence of the numerical error for smooth solutions. Numerical experiments for the 3D elastic wave equation in complex geometries corroborate the theoretical analysis. Numerical simulations of the 3D elastic wave equation in heterogeneous media with complex non-planar free surface topography are given, including numerical simulations of community developed seismological benchmark problems. Computational results show that even-order dual pair SBP FD operators are more efficient, robust and less prone to numerical dispersion errors on marginally resolved meshes when compared to the odd-order dual pair and traditional SBP FD operators. Finally, scaling tests demonstrate nearly perfect strong scaling and verify the efficiency of our parallel implementation of the high order upwind SBP methods.

Spreading Speeds of Nonlocal KPP Equations in Heterogeneous Media

This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media. The kernel K is assumed to depend on the media. First, we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues. Second, we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.

Carleman estimates for the wave equation in heterogeneous media with non-convex interface

A wave equation whose main coefficient is discontinuous models the evolution of waves amplitude in a media composed of at least two different materials, in which the propagation speed is different. In our mathematical setting, the spatial domain where the partial differential equation evolves is an open bounded subset of R2 and the wave speed is assumed to be constant in each one of two sub-domains, separated by a smooth and possibly non-convex interface. This article is concerned with the construction of Carleman weights for this wave operator, allowing generalizations of previous results to the case of an interface that is not necessarily the boundary of a convex set. Indeed, using the orthogonal projection onto this interface, we define convex functions satisfying the transmission conditions imposed by the equation, such that, under usual hypothesis on the sign of the jump of the wave speed, can be used as Carleman weights.

Optimization of Bathymetry for Long Waves with Small Amplitude

This paper deals with bathymetry-oriented optimization in the case of long waves with small amplitude. Under these two assumptions, the free-surface incompressible Navier-Stokes system can be written as a wave equation where the bathymetry appears as a parameter in the spatial operator. Looking then for time-harmonic fields and writing the bottom topography as a perturbation of a flat bottom, we end up with a heterogeneous Helmholtz equation with impedance boundary condition. In this way, we study some PDE-constrained optimization problem for a Helmholtz equation in heterogeneous media whose coefficients are only bounded with bounded variation. We provide necessary condition for a general cost function to have at least one optimal solution. We also prove the convergence of a finite element approximation of the solution to the considered Helmholtz equation as well as the convergence of discrete optimum toward the continuous ones. We end this paper with some numerical experiments to illustrate the theoretical results and show that some of their assumptions could actually be removed.

The heterogeneous Helmholtz problem with spherical symmetry: Green’s operator and stability estimates

We study wave propagation phenomena modelled in the frequency domain by the Helmholtz equation in heterogeneous media with focus on media with discontinuous, highly oscillating wave speed. We restrict to problems with spherical symmetry and will derive explicit representations of the Green’s operator and stability estimates which are explicit in the frequency and the wave speed.

On block triangular preconditioned iteration methods for solving the Helmholtz equation

To further improve the efficiency of solving the Helmholtz equation in heterogeneous media with large wavenumber, the Krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two-by-two linear system derived from the discrete Helmholtz equation. We further develop the eigenvalue properties of the preconditioned matrices to discuss the convergence of the corresponding preconditioned iteration methods. The superiority of such preconditioned iteration methods is prominent according to the numerical results when comparing with other classical iteration methods. We also investigate how the wavenumber influences the performance of the corresponding methods and it is shown that the iteration number of our proposed methods linearly increase with the wavenumber, roughly. Furthermore, the computational wave-fields which conform the real physical law are exhibited to show the correctness of our proposed numerical modeling algorithm.

Space-Time Discontinuous Petrov–Galerkin Methods for Linear Wave Equations in Heterogeneous Media

Abstract We establish an abstract space-time DPG framework for the approximation of linear waves in heterogeneous media. The estimates are based on a suitable variational setting in the energy space. The analysis combines the approaches for acoustic waves of Gopalakrishnan–Sepulveda [J. Gopalakrishnan and P. Sepulveda, A space-time DPG method for the wave equation in multiple dimensions, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 129–154] and Ernesti–Wieners [J. Ernesti and C. Wieners, A space-time discontinuous Petrov–Galerkin method for acoustic waves, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 99–127] and is based on the abstract definition of traces on the skeleton of the time-space substructuring. The method is evaluated by large-scale parallel computations motivated from applications in seismic imaging, where the computational domain can be restricted substantially to a subset of the full space-time cylinder.

Solutions for a fractional diffusion equation in heterogeneous media

We investigate diffusive phenomena governed by fractional diffusion equations in the presence of a spatial dependent diffusion coefficient and external forces. We consider a fractional operator which enables us to analyze, in a unified way, different types of fractional time derivatives. This approach opens the possibility of considering new situations related to memory effects. After determining exact analytical solutions, we show that different diffusive behaviors can be obtained, depending on the fractional time operator employed. In this framework, we also discuss the role of the anomalous diffusion processes.

Dynamic data-driven Bayesian GMsFEM

In this paper, we propose a Bayesian approach for multiscale problems with the availability of dynamic observational data. Our method selects important degrees of freedom probabilistically in a Generalized multiscale finite element method framework. Due to scale disparity in many multiscale applications, computational models cannot resolve all scales. Dominant modes in the Generalized Multiscale Finite Element Method are used as “permanent” basis functions, which we use to compute an inexpensive multiscale solution and the associated uncertainties. Through our Bayesian framework, we can model approximate solutions by selecting the unresolved scales probabilistically. We consider parabolic equations in heterogeneous media. The temporal domain is partitioned into subintervals. Using residual information and given dynamic data, we design appropriate prior distribution for modeling missing subgrid information. The likelihood is designed to minimize the residual in the underlying PDE problem and the mismatch of observational data. Using the resultant posterior distribution, the sampling process identifies important degrees of freedom beyond permanent basis functions. The method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations.

Stability of High-Order Staggered-Grid Schemes for 3D Elastic Wave Equation in Heterogeneous Media

In this paper, we firstly derive the stability conditions of high-order staggered-grid schemes for the three-dimensional (3D) elastic wave equation in heterogeneous media based on the energy method. Moreover, the plane wave analysis yields a sufficient and necessary stability condition by the von Neumann criterion in homogeneous case. Numerical computations for 3D wave simulation with point source excitation are given.