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 develop and analyze a rigorous multiscale upscaling method for dual continuum model, which serves as a powerful tool in subsurface formation applications. Our proposed method is capable of identifying different continua and capturing non-local transfer and effective properties in the computational domain via constructing localized multiscale basis functions. The construction of the basis functions consists of solving local problems defined on oversampling computational region, subject to the energy minimizing constraints that the mean values of the local solution are zero in …
Skin contraction is an important biophysical process that takes place during and after the recovery of deep tissue injury. This process is mainly caused by fibroblasts (skin cells) and myofibroblasts (differentiated fibroblasts) that exert pulling forces on the surrounding extracellular matrix (ECM). Modelling is done in multiple scales: agent-based modelling on the microscale and continuum-based modelling on the macroscale. In this manuscript, we present some results from our study of the connection between these scales. For the one-dimensional case, we managed to rigorously establish the lin…
We will develop a nonlinear upscaling method for nonlinear transport equation. The proposed scheme gives a coarse scale equation for the cell average of the solution. In order to compute the parameters in the coarse scale equation, a local downscaling operator is constructed. This downscaling operation recovers fine scale properties using cell averages. This is achieved by solving the equation on an oversampling region with the given cell average as constraint. Due to the nonlinearity, one needs to compute these downscaling operations on the fly and cannot pre-compute these quantities. In orde…
Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using Non-Local Multi-Continuum method (NLMC)
Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；The Chinese University of Hong Kong（香港中文大学）
In this work, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. We consider unsaturated and two-phase flow problems in heterogeneous and fractured porous media, where mathematical models are formulated as general multicontinuum models. We construct a fine grid approximation using the finite volume method and embedded discrete fracture model. Macroscopic models for these complex nonlinear systems require nonlocal multicontinua approaches, which are developed in earlier works. These rigorous techniques require complex local computations, which in…
Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media
Institute for Scientific Computation, Texas A&M University（德州农工大学科学计算研究所）；Department of Computational Technologies, North-Eastern Federal University（东北联邦大学计算技术系）；Department of Mathematics, The Chinese University of Hong Kong (CUHK)（香港中文大学数学系）；Department of Mathematic, Texas A&M University（德州农工大学数学系）；Multiscale model reduction laboratory, North-Eastern Federal University（东北联邦大学多尺度模型降维实验室）
In this study, we consider the nonlinear Schödinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean space with simultaneous reduction in the lattice distance and expansion of the domain. Moreover, we obtain a precise global-in-time bound for the rate of convergence. Our proof heavily relies on Strichartz estimates on a finite lattice. A key observation is that, compared to the case of a lattice with a fixed size [Y. Hong, C. Kwak, S. Nakamura, and C. Yang, …
A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first order hyperbolic formulation of continuum fluid and solid mechanics. The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. Furthermore, the GCL is satisfied by a compatible discretization that makes use of a nodal solver to compute vertex-based fluxes that are used both for the motion of the computational mesh as well as for the time evolution of the governing PDEs. Second-order …
In fluid flow simulation, the multi-continuum model is a useful strategy. When the heterogeneity and contrast of coefficients are high, the system becomes multiscale, and some kinds of reduced-order methods are demanded. Combining these techniques with nonlinearity, we will consider in this paper a dual-continuum model which is generalized as a multi-continuum model for a coupled system of nonlinear Richards equations as unsaturated flows, in complex heterogeneous fractured porous media; and we will solve it by a novel multiscale approach utilizing the constraint energy minimizing generalized …
