Results in Applied Mathematics最新文献

In this article, a fourth order compact difference scheme for solving the two-dimensional extended Fisher–Kolmogorov (2D EFK) equation is proposed and analyzed. This scheme is three-level implicit, based on a novel time discretization idea of $u(x_i,y_j,t_n)\approx \frac{1}{4}(U_{i,j}^{n+1}+2U_{i,j}^n+U_{i,j}^{n-1})$. The discrete energy functional method is used to obtain prior estimates of numerical solutions in the maximum norm. Furthermore, the convergence of the difference solutions in the maximum norm is analyzed, and the convergence rate is obtained as $O({\tau }^2+h^4)$, which without any restriction on the grid ratio with time step $\tau$ and mesh size $h$. Finally, numerical examples are given to support the theoretical analysis.

In this paper we consider computational challenges associated with thermo-hydro-mechanical models for simulation of subsidence due to permafrost thaw. The model we outline couples heat conduction with phase change and thermal advection to Biot’s poroelasticity equations with attention paid to the dependence of the constitutive parameters on temperature. Our numerical scheme uses the lowest order mixed finite elements for discretization of thermal and hydrological flow, and Galerkin finite elements for mechanics, and uses an implicit–explicit time stepping. We set up an iterative solver that solves the thermal subproblem followed by the hydro-mechanical subproblem, and demonstrate its robustness in practical heterogeneous permafrost scenarios. We also identify the challenges associated with the roughness of the dependence of mechanical parameters on the temperature.

We propose four-field and five-field Hu–Washizu-type mixed formulations for nonlinear poroelasticity – a coupled fluid diffusion and solid deformation process – considering that the permeability depends on a linear combination between fluid pressure and dilation. As the determination of the physical strains is necessary, the first formulation is written in terms of the primal unknowns of solid displacement and pore fluid pressure as well as the poroelastic stress and the infinitesimal strain, and it considers strongly symmetric Cauchy stresses. The second formulation imposes stress symmetry in a weak sense and it requires the additional unknown of solid rotation tensor. We study the unique solvability of the problem using the Banach fixed-point theory, properties of twofold saddle-point problems, and the Banach–Nečas–Babuška theory. We propose monolithic Galerkin discretisations based on conforming Arnold–Winther for poroelastic stress and displacement, and either PEERS or Arnold–Falk–Winther finite element families for the stress–displacement-rotation field variables. The wellposedness of the discrete problem is established as well, and we show a priori error estimates in the natural norms. Some numerical examples are provided to confirm the rates of convergence predicted by the theory, and we also illustrate the use of the formulation in some typical tests in Biot poroelasticity.

We make an experimental comparison of methods for computing the numerical radius of an $n\times n$ complex matrix, based on two well-known characterizations, the first a nonconvex optimization problem in one real variable and the second a convex optimization problem in $n^2+1$ real variables. We make comparisons with respect to both accuracy and computation time using publicly available software.

The rimless wheel model appears in engineering literature as a zero dynamics reduction of the model of a biped walking down a slope. Nonuniformity in the rimless wheel (i.e. unequal lengths of spokes and unequal angles between successive spokes) can be viewed as nonuniformity of the walking terrain. Existence of a limit cycle for nonuniform rimless wheel model was established in the earlier literature but stability was addressed just briefly. The present paper formulates conditions for asymptotic stability of the limit cycle and verifies the findings with numeric simulations (Wolfram Mathematica notebook is uploaded with this paper as supplementary material).

This proposed work concerns the nonlocal controllability criteria for state delay with an impulsive fractional integro-differential equation in n-dimensional Euclidean space in the sense of the Caputo fractional derivative. The mild solution is attained through the standard Laplace transform and iterative process. In particular, we obtained sufficient conditions by using degree theory. In addition, we exhibit the unique solution and nonlocal controllability criteria of our given problem through Gronwall’s inequality and appropriate assumptions. At last, we examine the precision of our findings using numerical computations and applications of the adaptive framework we have provided.

To study the influence of environmental pollutants on the dynamics of river populations, we present a process-oriented model that elucidates the interplay between a population and toxic substances in a flowing environment. Here, an unknown boundary defines the front for the spread of the toxicant in the aquatic environment. Some properties of the solution are studied, and the global solvability of the problem is established. To solve problems with a free boundary, we establish Schauder-type estimates. The behaviour of a free boundary has been studied.

Numerous studies have been conducted to investigate porous systems under different damping effects. Recent research has consistently achieved the expected exponential decay of energy solutions when employing stabilization techniques that involve non-physical assumptions of equal wave velocities. In this study, we examine a one-dimensional thermoelastic porous system within the framework of the second frequency spectrum. Remarkably, we demonstrate that exponential decay can be achieved without relying on the assumption of equal wave speeds. We consider the porous system, and we incorporated thermoelastic damping based on the Green–Naghdi law of heat conduction into our study. To begin with, we use the Faedo–Galerkin approximation method to validate the global well-posedness of the system. By utilizing a Lyapunov functional, we establish exponential stability without relying on the assumption of equal wave speed. We then introduce and analyze a numerical scheme. Finally, by assuming additional regularity of the solution, we derive a priori error estimates.

Multiphysics processes in fractured porous media is a research field of importance for several subsurface applications and has received considerable attention over the last decade. The dynamics are characterized by strong couplings between processes as well as interaction between the processes and the structure of the fractured medium itself. The rich range of behaviour calls for explorative mathematical modelling, such as experimentation with constitutive laws and novel coupling concepts between physical processes. Moreover, efficient simulations of the strong couplings between multiphysics processes and geological structures require the development of tailored numerical methods.

We present a modelling framework and its implementation in the open-source simulation toolbox PorePy, which is designed for rapid prototyping of multiphysics processes in fractured porous media. PorePy uses a mixed-dimensional representation of the fracture geometry and generally applies fully implicit couplings between processes. The code design follows the paradigms of modularity and differentiable programming, which together allow for extreme flexibility in experimentation with governing equations with minimal changes to the code base. The code integrity is supported by a multilevel testing framework ensuring the reliability of the code.

We present our modelling framework within a context of thermo-poroelasticity in deformable fractured porous media, illustrating the close relation between the governing equations and the source code. We furthermore discuss the design of the testing framework and present simulations showcasing the extendibility of PorePy, as well as the type of results that can be produced by mixed-dimensional simulation tools.

We study the existence, uniqueness and stability of the steady state for the dynamic described by a class of first-order difference equations. We then apply the result to analyse a housing market where the supply is linear and demand is a bounded and monotone decreasing function of price, derived from households’ optimization behaivour. Under two linear price adjustment mechanisms, we prove the existence and uniqueness of an equilibrium, which is independent of the mechanisms. That is, the house price converges to a same steady state where it clears the market under both mechanisms. The result is general in the sense that we do not need to specify a particular form of demand function. Besides, the same approach can be utilized to analyse the dynamics of other markets.