This work continues a line of works on developing partially explicit methods for multiscale problems. In our previous works, we have considered linear multiscale problems, where the spatial heterogeneities are at subgrid level and are not resolved. I
n these works, we have introduced contrast-independent partially explicit time discretizations for linear equations. The contrast-independent partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting is proposed that treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that it guarantees a stability and formulate a condition for the time stepping. This condition is formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.
We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The
resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract the directions, which are further incorporated into a ray-based interior-penalty discontinuous Galerkin (IPDG) method to solve the Helmholtz equations at higher frequencies. In this way, we observe no apparent pollution effects in the resulting Helmholtz solutions in inhomogeneous media. Our 2D and 3D numerical results show that the proposed scheme is very efficient and yields highly accurate solutions.
In this work, we design and investigate contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. We consider multiscale problems, where the spatial heterogeneities are at subgrid level and
are not resolved. In our previous work, we have introduced contrast-independent partially explicit time discretizations and applied to parabolic equations. The main idea of contrast-independent partially explicit time discretization is to split the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Using this decomposition, our goal is further appropriately to introduce time splitting such that the resulting scheme is stable and can guarantee contrast-independent discretization under some suitable (reasonable) conditions. In this paper, we propose contrast-independent partially explicitly scheme for wave equations. The splitting requires a careful design. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (slow). This condition requires some type of non-contrast dependent space and is easier to satisfy in the slow space. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.
Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These is
sues to some extend are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is $dt=H^2/kappa_{max}$, where $kappa_{max}$ is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The unconditional stability of these algorithms require a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.
In this work, we propose a local multiscale model reduction approach for the time-domain scalar wave equation in a heterogenous media. A fine mesh is used to capture the heterogeneities of the coefficient field, and the equation is solved globally on
a coarse mesh in the discontinuous Galerkin discretization setting. The main idea of the model reduction approach is to extract dominant modes in local spectral problems for representation of important features, construct multiscale basis functions in coarse oversampled regions by constraint energy minimization problems, and perform a Petrov-Galerkin projection and a symmetrization onto the coarse grid. The method is expicit and energy conserving, and exhibits both coarse-mesh and spectral convergence, provided that the oversampling size is appropriately chosen. We study the stability and convergence of our method. We also present numerical results on the Marmousi model in order to test the performance of the method and verify the theoretical results.
Recently, several approaches for multiscale simulations for problems with high contrast and no scale separation are introduced. Among them is the nonlocal multicontinua (NLMC) method, which introduces multiple macroscopic variables in each computatio
nal grid. These approaches explore the entire coarse block resolution and one can obtain optimal convergence results independent of contrast and scales. However, these approaches are not amenable to many multiscale simulations, where the subgrid effects are much smaller than the coarse-mesh resolution. For example, the molecular dynamics of shale gas occurs in much smaller length scales compared to the coarse-mesh size, which is of orders of meters. In this case, one can not explore the entire coarse-grid resolution in evaluating effective properties. In this paper, we merge the concepts of nonlocal multicontinua methods and Representative Volume Element (RVE) concepts to explore problems with extreme scale separation. The first step of this approach is to use sub-grid scale (sub to RVE) to write a large-scale macroscopic system. We call it intermediate scale macroscale system. In the next step, we couple this intermediate macroscale system to the simulation grid model, which are used in simulations. This is done using RVE concepts, where we relate intermediate macroscale variables to the macroscale variables defined on our simulation coarse grid. Our intermediate coarse model allows formulating macroscale variables correctly and coupling them to the simulation grid. We present the general concept of our approach and present details of single-phase flow. Some numerical results are presented. For nonlinear examples, we use machine learning techniques to compute macroscale parameters.
Numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas. However, numerical solutions on the fine grid are often prohibitively expensive, and multiscale model reduction te
chniques are introduced to efficiently solve for an accurate approximation on the coarse grid. In this paper, we propose an energy minimization based multiscale model reduction approach in the discontinuous Galerkin discretization setting. The main idea of the method is to extract the non-decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems, and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems. The multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous Galerkin formulation. The minimal degree of freedom in representing high-contrast features is achieved through the design of local spectral problems, which provides the most compressed local multiscale space. We analyze the method for solving Darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast, provided that the oversampling size is appropriately chosen. Numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high-contrast heterogeneous media.
In this paper, we develop a space-time upscaling framework that can be used for many challenging porous media applications without scale separation and high contrast. Our main focus is on nonlinear differential equations with multiscale coefficients.
The framework is built on nonlinear nonlocal multi-continuum upscaling concept and significantly extends the results in the proceeding paper. Our approach starts with a coarse space-time partition and identifies test functions for each partition, which plays a role of multi-continua. The test functions are defined via optimization and play a crucial role in nonlinear upscaling. In the second stage, we solve nonlinear local problems in oversampled regions with some constraints defined via test functions. These local solutions define a nonlinear map from macroscopic variables determined with the help of test functions to the fine-grid fields. This map can be thought as a downscaled map from macroscopic variables to the fine-grid solution. In the final stage, we seek macroscopic variables in the entire domain such that the downscaled field solves the global problem in a weak sense defined using the test functions. We present an analysis of our approach for an example nonlinear problem. Our unified framework plays an important role in designing various upscaled methods. Because local problems are directly related to the fine-grid problems, it simplifies the process of finding local solutions with appropriate constraints. Using machine learning (ML), we identify the complex map from macroscopic variables to fine-grid solution. We present numerical results for several porous media applications, including two-phase flow and transport.
In this paper, we discuss multiscale methods for nonlinear problems. The main idea of these approaches is to use local constraints and solve problems in oversampled regions for constructing macroscopic equations. These techniques are intended for pro
blems without scale separation and high contrast, which often occur in applications. For linear problems, the local solutions with constraints are used as basis functions. This technique is called Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). GMsFEM identifies macroscopic quantities based on rigorous analysis. In corresponding upscaling methods, the multiscale basis functions are selected such that the degrees of freedom have physical meanings, such as averages of the solution on each continuum. This paper extends the linear concepts to nonlinear problems, where the local problems are nonlinear. The main concept consists of: (1) identifying macroscopic quantities; (2) constructing appropriate oversampled local problems with coarse-grid constraints; (3) formulating macroscopic equations. We consider two types of approaches. In the first approach, the solutions of local problems are used as basis functions (in a linear fashion) to solve nonlinear problems. This approach is simple to implement; however, it lacks the nonlinear interpolation, which we present in our second approach. In this approach, the local solutions are used as a nonlinear forward map from local averages (constraints) of the solution in oversampling region. This local fine-grid solution is further used to formulate the coarse-grid problem. Both approaches are discussed on several examples and applied to single-phase and two-phase flow problems, which are challenging because of convection-dominated nature of the concentration equation.
The dual continuum model serves as a powerful tool in the modeling of subsurface applications. It allows a systematic coupling of various components of the solutions. The system is of multiscale nature as it involves high heterogeneous and high contr
ast coefficients. To numerically compute the solutions, some types of reduced order methods are necessary. We will develop and analyze a novel multiscale method based on the recent advances in multiscale finite element methods. Our method will compute multiple local multiscale basis functions per coarse region. The idea is based on some local spectral problems, which are important to identify high contrast channels, and an energy minimization principle. Using these concepts, we show that the basis functions are localized, even in the presence of high contrast long channels and fractures. In addition, we show that the convergence of the method depends only on the coarse mesh size. Finally, we present several numerical tests to show the performance.
