高等学校計算数学学報 = Numerical mathematics

dynamics. We have arrived at an interesting concept, seeing solutions as continuous curves moving around in an infinite-dimensional metric space X (here, the function space L1(Ω)). Viewing solutions as continuous curves in a general space is the starting point of the abstract theory of differential equations, a way that we will travel quite often. In the so-called Abstract Dynamics it is typical to forget the variable x in the notation and look at the map t 7→ u(t) ∈ X, where u(t) is the abbreviated form for u(·, t). Remarks. (1) Note that the theorem allows to define the value u(t) of a limit solution (in particular, of a weak solution) u at any time t > 0 as a well-defined element of L1(Ω). Actually, in many cases, as when Φ is superlinear and f is bounded, it is an element of L∞(Ω). (2) If u0 and f are bounded the initial regularity is better. In that case the initial data are taken in the Lp sense: ũ(t) → ũ(0) in Lp(Ω), for every p 0; if u0 is continuous, then the convergence takes place uniformly in x as t → 0, see Section 7.5.1. (3) Unfortunately, there are no equivalent L1 estimates for the Dirichlet Problem with nonhomogeneous data g 6= 0. We end this subsection with a simple but very useful consequence. Corollary 6.3 Let u be a limit solution with data u0 ∈ L1(Ω) and f ∈ L1(Q). If t1 > 0, then ũ(x, t) = u(x, t + t1) is the limit solution with data ũ0(x) = u(x, t1) and forcing term f(x, t) = f(x, t + t1). This important result is immediate for the approximations. We leave the details to the reader. Remark. Let us note that any concept of limit solution depends on the type of admissible approximations and on the functional setting in which limits are taken. The definition we propose applies in the L1 setting. If needed, these solutions will be called L1-limit solutions. For an extension see Section 6.6. 6.2 Theory of very weak solutions The continuous dependence with respect to the L1-norm is a powerful property. It has allowed us to extend the existence result for weak solutions of the preceding section and

The Porous Medium Equation

An inexact Newton method is a generalization of Newton’s method for solving $F(x) = 0,F:\mathbb{R}^n \to \mathbb{R}^n $in which, at the kth iteration, the step $s_k $ from the current approximate solution $x_k $ is required to satisfy a condition $\|F(x_k ) + F'(x_k )s_k \| \leqslant \eta _k \|F(x_k )\|$ for a “forcing term” $\eta _k \in [0,1)$. In typical applications, the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well. Promising choices of the forcing terms are given, their local convergence properties are analyzed, and their practical performance is shown on a representative set of test problems.

Choosing the forcing terms in an inexact Newton method

A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.

/pdf/front-propagation-in-heterogeneous-media-4gz5orzuvv.pdf

Front Propagation in Heterogeneous Media

Variable-density flow and transport in porous media: approaches and challenges

We discuss an extension of the Buckley–Leverett (BL) equation describing two-phase flow in porous media. This extension includes a third order mixed derivatives term and models the dynamic effects in the pressure difference between the two phases. We derive existence conditions for traveling wave solutions of the extended model. This leads to admissible shocks for the original BL equation, which violate the Oleinik entropy condition and are therefore called nonclassical. In this way we obtain nonmonotone weak solutions of the initial-boundary value problem for the BL equation consisting of constant states separated by shocks, confirming results obtained experimentally.

/pdf/a-new-class-of-entropy-solutions-of-the-buckley-leverett-2kynrhc2zt.pdf

A new class of entropy solutions of the Buckley-Leverett equation

We consider a model for non-static groundwater flow where the saturation-pressure relation is extended by a dynamic term. This approach, together with a convective term due to gravity, results in a pseudo-parabolic Burgers type equation. We give a rigorous study of global travelling-wave solutions, with emphasis on the role played by the dynamic term and the appearance of fronts.

/pdf/infiltration-in-porous-media-with-dynamic-capillary-pressure-2rx6vqva63.pdf

Infiltration in porous media with dynamic capillary pressure: travelling waves

We study travelling wave solutions of a model describing solute transport through porous media, where the chemical species undergoes absorption, i.e. a retention/release reaction with the surface of the porous sceleton. Processes of this type are of fundamental importance in different fields. We do not consider the process at the microscopic scale of single grains and pores, but at the microscopic scale appropriate for the measurement of the phenomena described above

Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption : travelling waves

Similarity solution for capillary redistribution of two phases in a porous medium with a single discontinuity

In this paper we present a model for crystal dissolution in porous media and analyse travelling wave solutions of the ensuing equations for a one-dimensional flow situation. We demonstrate the structure of the waves and we prove existence and uniqueness. The travelling wave description is characterized by a rate parameter k and a diffusion / dispersion parameter D. We investigate the limit processes as k -> 8 and D -> 0 and we obtain expressions for the rate of convergence. We also present some numerical results.

van Cj Hans Duijn

Papers

A new class of entropy solutions of the Buckley-Leverett equation

Infiltration in porous media with dynamic capillary pressure: travelling waves

Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption : travelling waves

Similarity solution for capillary redistribution of two phases in a porous medium with a single discontinuity

Travelling wave behaviour of crystal dissolution in porous media flow