Porous Medium Equation Research Articles

Homogenization of Richards' equations in multiscale porous media with soft inclusions

The paper is devoted to the homogenization of Richards' type equations in a deterministic multiscale porous medium filled with soft inclusions. The medium consists of a deterministic ensemble of coarse aggregates embedded within a matrix containing a deterministic ensemble of fine aggregates, leading to a network in which the inclusions (of different sizes) are widely distributed. Under various settings involving the structure of the medium and the coefficients of the equation, we obtain homogenized limits of the underlying micro-model. We also provide a suitable approximation scheme for the homogenized coefficients, thereby providing a numerical method for computing these coefficients. To achieve our goal, apply tools of the multiscale sigma-convergence method.

The application of successive overrelaxation method for the solution of linearized half-sweep finite difference approximation to two-dimensional porous medium equation

Successive overrelaxation or S.O.R. method is a widely known parameter-based iterative method that can regulate a large and sparse system of equations so that the number of iterations required to solve the system can be reduced. Many researchers have applied the S.O.R. method to get the solution to the mathematical problem efficiently. This paper extends the application of the S.O.R. method to solve one of the nonlinear partial differential equation, which is the two-dimensional porous medium equation. The S.O.R. method is incorporated into an iterative method that is formulated based on a half-sweep finite difference approximation, and the Newton-type linearization solves the nonlinear term that presents in the equation. The numerical experiment that uses this innovative numerical method to solve several two-dimensional porous medium equation problems shows significant improvement to the percentage of reduction in the number of iterations and computation time.

Generalized semi-analytical solution for coupled multispecies advection-dispersion equations in multilayer porous media

Multispecies contaminant transport in the Earth’s subsurface is commonly modelled using advection-dispersion equations coupled via first-order reactions. Analytical and semi-analytical solutions for such problems are highly sought after but currently limited to either one species, homogeneous media, certain reaction networks, specific boundary conditions or a combination thereof. In this paper, we develop a semi-analytical solution for the case of a heterogeneous layered medium and a general first-order reaction network. Our approach combines a transformation method to decouple the multispecies equations with a recently developed semi-analytical solution for the single-species advection-dispersion-reaction equation in layered media. The generalized solution is valid for arbitrary numbers of species and layers, general Robin-type conditions at the inlet and outlet and accommodates both distinct retardation factors across layers or distinct retardation factors across species. Four test cases are presented to demonstrate the solution approach with the reported results in agreement with previously published results and numerical results obtained via finite volume discretisation. MATLAB code implementing the generalized semi-analytical solution is made available.

Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic p-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the “ess lim inf” of past times. We also show that it can be recovered by the limit of certain integral averages of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle.

Transport distances for PDEs: the coupling method

We informally reviewa fewPDEs for which some transport cost between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker–Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This means that we double the variables and solve, globally in time, a well-chosen PDE posed on the Euclidean square of the physical space. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge–Kantorovich problem.

Numerical investigation of the two-dimensional space-time fractional diffusion equation in porous media

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order $${\mathcal {O}}({\delta \tau }^{2-\alpha }),~ 0<\alpha <1$$ . Then, the Chebyshev spectral collocation of the third kind is implemented to approximate spatial variables and to obtain full discretization of the equation. Moreover, the unconditional stability and convergence of the proposed method are shown in the perspective $$H^{2}$$ -norm. Two numerical examples are presented to verify the effectiveness and the accuracy of the proposed method. The comparison between our obtained numerical results and the results of existing schemes in the literature shows that the proposed method is more reliable and precise.

Lamination Convex Hull of Stationary Incompressible Porous Media Equations

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 17 June 2020Accepted: 30 October 2020Published online: 21 January 2021Keywordsincompressible porous media equations, relaxation, lamination convex hullAMS Subject Headings35Q35, 76S05, 76B03Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Structure-Preserving Numerical Methods for Nonlinear Fokker--Planck Equations with Nonlocal Interactions by an Energetic Variational Approach

In this work, we develop novel structure-preserving numerical schemes for a class of nonlinear Fokker--Planck equations with nonlocal interactions. Such equations can cover many cases of importance, such as porous medium equations with external potentials, optimal transport problems, and aggregation-diffusion models. Based on the energetic variational approach, a trajectory equation is first derived by using the balance between the maximal dissipation principle and the least action principle. By a convex splitting technique, we propose energy dissipating numerical schemes for the trajectory equation. Rigorous numerical analysis reveals that the nonlinear numerical schemes are uniquely solvable, naturally respect mass conservation and positivity at the fully discrete level, and preserve steady states in an admissible convex set, where the discrete Jacobian of flow maps is positive. Under certain assumptions on smoothness and a positive Jacobian, the numerical schemes are shown to be second order accurate in space and first order accurate in time. Extensive numerical simulations are performed to demonstrate several valuable features of the proposed schemes. In addition to the preservation of physical structures, such as positivity, mass conservation, discrete energy dissipation, and steady states, numerical simulations further reveal that our numerical schemes are capable of solving degenerate cases of the Fokker--Planck equations effectively and robustly. It is shown that the developed numerical schemes have convergence order even in degenerate cases with the presence of solutions having compact support and can accurately and robustly compute the waiting time of free boundaries without any oscillation. The limitation of numerical schemes due to a singular Jacobian of the flow map is also discussed.

$L^1$-Convergence to Generalized Barenblatt Solution for Compressible Euler Equations with Time-Dependent Damping

The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(\rho)=\kappa\rho^{\gamma}, \gamma>1$) with time dependent damping like $-\frac{1}{(1+t)^\lambda}\rho u$ ($0<\lambda<1$) is investigated. By introducing an elaborate iterative method and using the intensive entropy analysis, it is proved that the $L^\infty$ entropy solution of compressible Euler equations with finite initial mass converges strongly in the natural $L^1$ topology to a fundamental solution of porous media equation (PME) with time-dependent diffusion, called by generalized Barenblatt solution. It is interesting that the $L^1$ decay rate is getting faster and faster as $\lambda$ increases in $(0, \frac{\gamma}{\gamma+2}]$, while is getting slower and slower in $[ \frac{\gamma}{\gamma+2}, 1)$.

The Waiting Time Phenomenon in Spatially Discretized Porous Medium and Thin Film Equations

Various degenerate diffusion equations exhibit a waiting time phenomenon: depending on the “flatness” of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin film equations, and we obtain sufficient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. For the spatially discrete solution, the waiting time phenomenon refers to a deviation of the edge of support from its original position by a quantity comparable to the mesh width, over a mesh-independent time interval. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique à la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.

The mathematical model of gas flowing in porous medium based on the homogenization method

Considered the characteristics of porous medium in the coal seam and goaf, in order to reflect the accurately influence of various porous media against the gas flow, the mathematical model of discrete multi-scale network and macroscopic flow, CFCM (Coal-Fracture-Cavity-Model), was presented. The porous medium is classified into coal matrix, fracture and hole systems based on the size, and the coal matrix system includes micro fractures and micro-porous. The coal matrix system and fracture system can be regarded as diffusion and percolation areas; hole system can be regarded as a free-flowing area. The computation model of flow field in micro-scale, small-scale and large-scale are obtained according the Fick’s diffusion law, Darcy’s permeability law and Forchheimer generalized Darcy law respectively, the homogenization method is used to analyse the mathematical model by scale upgrading and the equivalent Darcy’s fluid equation of porous medium is got to describe the characteristics of the medium in the flow field accurately. An example calculated shows that the coal matrix and fracture systems are the most influential factors of the flow field in goaf and the two systems above would prevent the diffusion of airflow. The study validates the correctness of the classification method and the model of flow equation.

Blow up analysis for a porous media equation with nonlinear sink and nonlinear boundary condition

Blow up analysis for a porous media equation with nonlinear sink and nonlinear boundary condition

Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients

&lt;p style='text-indent:20px;'&gt;In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions, which is interesting in its own right. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.&lt;/p&gt;

Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order

We prove well-posedness and regularity of solutions to a fractional diffusion porous media equation with a variable fractional order that may depend on the unknown solution. We present a linearly implicit time-stepping method to linearize and discretize the equation in time, and present rigorous analysis for the convergence of numerical solutions based on proved regularity results.

NUMERICAL SOLUTIONS OF THE NONLINEAR POROUS MEDIA EQUATION BASED ON HIGH EXPLORATION PARTICLE SWARM OPTIMIZATION AND MOVING LEAST SQUARES

In this study, a new numerical method based on the combination of High Exploration Particle Swarm Optimization (HEPSO) and Moving Least Squares (MLS) is introduced to solve nonlinear porous media equations. The MLS scheme is employed to describe an appropriate discretized function, and the penalty method is implemented to convert the constrained problem into an unconstrained one via satisfying the initial conditions. The identified objective function is minimized by the HEPSO to find the approximated nodal values for the nonlinear porous media equation. In order to illustrate the effectiveness of the HEPSO, the optimization trajectories are compared with those of a Standard Particle Swarm Optimization (SPSO) algorithm. Moreover, comparisons are made between the exact solution and the introduced strategy to expose the accuracy, effectiveness and simplicity of the proposed method.

Optimal regularity in time and space for the porous medium equation

Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal regularity of the linear case.

Harnack estimates for a kind of porous medium equation

In this paper, we derive several differential Harnack estimates (also known as Li–Yau–Hamilton type estimates) for nonnegative solutions and positive solutions of the porous medium equation ft=ffxx+fx2+f2, which is a nonlinear parabolic equation. Our derivations rely on an idea related to the parabolic maximum principle. We prove these estimates by different methods. As the applications of the estimates, Harnack inequalities are also obtained.

On stochastic porous-medium equations with critical-growth conservative multiplicative noise

<p style='text-indent:20px;'>First, we prove existence, nonnegativity, and pathwise uniqueness of martingale solutions to stochastic porous-medium equations driven by conservative multiplicative power-law noise in the Ito-sense. We rely on an energy approach based on finite-element discretization in space, homogeneity arguments and stochastic compactness. Secondly, we use Monte-Carlo simulations to investigate the impact noise has on waiting times and on free-boundary propagation. We find strong evidence that noise on average significantly accelerates propagation and reduces the size of waiting times – changing in particular scaling laws for the size of waiting times.

An optimized compact reconstruction weighted essentially non‐oscillatory scheme for advection problems

AbstractThis paper presents an optimized compact reconstruction weighted essentially non‐oscillatory scheme without dissipation errors (OCRWENO‐LD) for solving advection problems. The construction procedure of this optimized scheme without dissipation errors is as follows: (1) We first design a high‐order compact difference scheme with four general weights connecting four low‐order compact stencils. The four general weights are determined by applying the Taylor series expansions. (2) These general weights are optimized to the new weights which are derived from the WENO concept and modified wavenumber approach. (3) No dissipation errors are found for the developed OCRWENO‐LD scheme through Fourier analysis. The proposed high‐resolution scheme demonstrates its capability in exhibiting high‐accuracy in smooth regions and avoiding numerical oscillation near discontinuities when simulating the wave equation, Burgers' equation, one‐dimensional Euler equation, porous medium equation, and convection–diffusion Buckley–Leverett equation.

Regularity of solutions of a fractional porous medium equation

This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely \partial_t u = { \nabla \cdot} \left(u \nabla(-\Delta)^{\frac{\alpha}{2}-1}u^{m-1} \right) where u:\mathbb{R}_+\times \mathbb{R}^N \to \mathbb{R}_+ , for 0 &lt; \alpha &lt; 2 and m\geq2 . We prove that the L^1\cap L^\infty weak solutions constructed by Biler, Imbert and Karch (2015) are locally Hölder-continuous in time and space. In this article, the classical parabolic De Giorgi techniques for the regularity of PDEs are tailored to fit this particular variant of the PME equation. In the spirit of the work of Caffarelli, Chan and Vasseur (2011), the two main ingredients are the derivation of local energy estimates and a so-called “intermediate value lemma”. For \alpha\leq1 , we adapt the proof of Caffarelli, Soria and Vázquez (2013), who treated the case of a linear pressure law. We then use a non-linear drift to cancel out the singular terms that would otherwise appear in the energy estimates.