Fourth-order Diffusion Equation Research Articles

A finite element method for stochastic diffusion equations using fluctuating hydrodynamics

We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing equations are stochastic partial differential equations with a fluctuating forcing term. We propose a discrete formulation of the fluctuating forcing term that has the correct covariance matrix up to a standard discretization error. Furthermore, we derive a linear mapping to transform the finite element solution into an equivalent discrete solution that is free of the artificial correlations introduced by the spatial discretization. The method is validated by applying it to two diffusion problems: a second-order diffusion equation and a fourth-order diffusion equation. The theoretical (continuum) solution to the first case presents spatially decorrelated fluctuations, while the second case presents fluctuations correlated over a finite length. In both cases, the numerical solution presents a structure factor that approximates well the continuum one.

Automatic Extraction of Martian Subsurface Layer from Radargrams Based on PDE Denoising and KL Filter

The polar regions of Mars, including the South and North Poles, are crucial for studying Martian climate and geological history, as they contain the largest reservoir of subsurface water ice. This study introduces a new approach for reflector detection, which includes radargram denoising to effectively enhance the signal of underground reflectors, peak detection to extract the positions of subsurface stratification from the radar echoes, and peak points connection to form continuous layers. The mapped enhancement denoising process involves a linear brightness adjustment and a fourth-order diffusion equation to enhance the signal of the subsurface layers for effective detection. The subsurface detection extracts the surface and subsurface peak points based on a peak detection algorithm, while using locally window-enhanced peak filtering and Kullback–Leibler (KL) divergence mapping to filter out non-stratified peak points. Finally, the layered connection process uses the proximity parameter to connect peak points in the same layer. Applied to multiple SHARAD (Shallow Radar) images at the Martian poles, this algorithm demonstrated a false detection rate below 5%. Compared to other methods, this method has a missed detection rate of less than 5% and, additionally, exhibits fewer discontinuities in layer connectivity. Therefore, this algorithm shows exceptional proficiency and applicability in analyzing the complex subsurface structures of the Martian polar regions.

Compact difference scheme for time-fractional nonlinear fourth-order diffusion equation with time delay

We construct a compact difference scheme for two-dimensional time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the linearized compact difference scheme is formulated by employing the L2−1σ formula and the compact difference operator to approximate temporal Caputo derivative and spatial second-order derivatives, respectively. The unique solvability of the proposed scheme is proved. Then the scheme is proved to be stable and convergent with the rate of second order in time and fourth order in space. The efficiency of the scheme is supported by some numerical experiments.

A new anisotropic fourth-order diffusion equation model based on image features for image denoising

<p style='text-indent:20px;'>Image denoising has always been a challenging task. For performing this task, one of the most effective methods is based on variational PDE. Inspired by the LLT model, we first propose a new adaptive LLT model by adding a weighted function, and then we propose a class of fourth-order diffusion equations based on the new functional. Owing to the adaptive function, the new functional is better than the LLT model and other fourth-order models in terms of edge preservation. While generalizing the Euler-Lagrange equation of the new functional, we discuss a new fourth-order diffusion framework for image denoising. Different from those of other fourth-order diffusion models, the new diffusion coefficients depend on the first-order and second-order derivatives, which can preserve edges and smooth images, respectively. Regarding numerical implementations, we first design an explicit scheme for the proposed model. However, fourth-order diffusion equations require strict stability conditions, and the number of iterations needed is considerable. Consequently, we apply the fast explicit diffusion algorithm (FED) to the explicit scheme to reduce the time consumption of the proposed approach. Furthermore, the additive operator splitting (AOS) scheme is applied for the numerical implementation, and it is the most efficient among all of our algorithms. Finally, compared with other models, the new model exhibits superior effectiveness and efficiency.</p>

Shock filter coupled with a high-order PDE for additive noise removal and image quality enhancement

We are interested in repairing blurry images degraded by additive noise. We present a new filter for image enhancement based on the combination of a shock filter with a fourth order diffusion equation. The algorithm that we propose eliminates not only the blurry but also the noise. The control of the diffusion weight and that of the enhancement of the model is done by a parameter. After experimentation on some test images and on some real images of degraded old documents, the results obtained are satisfactory compared to certain models in the literature.

Adaptive fourth-order diffusion smoothing via bilateral kernel

This paper presents a flexible and adaptive fourth-order diffusion model for image noise removal. A flexible fourth-order diffusion equation is first presented. Then, an efficient and robust contextual discontinuity indicator based on the bilateral kernel is constructed and combined with the local discontinuity indicator (local spatial gradient) to get the desired adaptive fourth-order diffusion model. The proposed model degenerates to an anisotropic equation which performs the tangent smoothing for sharpening the inter-object boundaries and reduces to an isotropic equation which implements the fast smoothing in the homogenous regions for noise removal. Moreover, it is able to overcome the staircase artifacts arisen by most of second-order diffusion models. Experimental results are provided to validate the effectiveness of the proposed model with regard to objective evaluation metrics and visual performance which outperform those of some benchmark models.

Are waves with negative spatial damping unstable?

Conventional plane harmonic waves decay in direction of propagation, but unconventional harmonic waves grow in the direction of propagation. While a single unconventional wave cannot be a solution to a physically meaningful boundary value problem, these waves may have an essential contribution to the overall solution of a problem as long as this is a superposition of unconventional and conventional waves. A fourth order diffusion equation with proper thermodynamic structure, and the Burnett equations of rarefied gas dynamics exhibit conventional and unconventional waves. Steady state oscillating boundary value problems are considered to discuss the interplay of conventional and unconventional waves. Results show that as long as the second law of thermodynamics is valid, unconventional waves may contribute to the overall solution, which, however is dominated by conventional waves, and behaves as these.

Green’s functions for the fourth-order diffusion equation

Distinct physical processes are involved in the representation of diffusion phenomenon. On the other side, the more commonly employed differential equation for diffusion in the reviewed literature does not provide neither complete nor necessarily adequate mathematical representation of such processes what ends up to still be an ongoing unsolved challenge. In particular, this complex task of mathematical modeling is more visible in when, for instance, a set of diffusive particles are characterized by the simultaneous existence of two states of energy, possessing both a deceleration and temporary retention of particles. The fundamental hypothesis considered in this paper is that there is a predominant flux, which is the one represented by the potential defined by means of the Fick’s law. In addition, after a short period, other possible excitation microstates begin to exist, leading to the modeling of such physical phenomena using the fourth-order anomalous diffusion equation. Given the exposed framework, the present work contributes with the development of the analytical solution of a partial differential equation (PDE) that represents the diffusion process caused by a pulse of local and instantaneous energy. Green’s function is obtained through the solution of a specific PDE associated with the equation governing the original problem and in the same domain with the support of Fourier and Laplace transforms. These expressions were then validated using the finite difference numerical method. The successful proposed approach allowed us to produce completely new results for the diffusion and propagation processes, also taking into account retention processes or retardation effects, propagation disorders, acceleration or deceleration, or even fluid stagnation.

A numerical method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation

In this paper, we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation. In order to avoid using higher order elements, we introduce an intermediate variable [Formula: see text] and translate the fourth-order derivative of the original problem into a second-order coupled system. We discretize the fractional time derivative terms by using the [Formula: see text]-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula. In the fully discrete scheme, we implement the finite element method for the spatial approximation. Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained. Numerical experiments are carried out to demonstrate our theoretical analysis.

Unconditionally stable compact theta schemes for solving the linear and semi-linear fourth-order diffusion equations

This paper is concerned with numerical methods for solving a class of fourth-order diffusion equations. Combining the compact difference operator in space discretization and the linear θ method in time, the compact theta scheme for the linear problem is first proposed. By virtue of the Fourier method, the suggested scheme is shown to be unconditionally stable and convergent in the discrete L2-norm for any θ ∈ [1/2, 1]. And then this idea is generalized to the semi-linear case, the corresponding compact theta scheme is constructed and analyzed in detail. Numerical experiments corresponding to the linear and semi-linear situations are carried out to support our theoretical statements.

Multi-scale Anisotropic Fourth-Order Diffusion Improves Ridge and Valley Localization

Ridge and valley enhancing filters are widely used in applications such as vessel detection in medical image computing. When images are degraded by noise or include vessels at different scales, such filters are an essential step for meaningful and stable vessel localization. In this work, we propose a novel multi-scale anisotropic fourth-order diffusion equation that allows us to smooth along vessels, while sharpening them in the orthogonal direction. The proposed filter uses a fourth order diffusion tensor whose eigentensors and eigenvalues are determined from the local Hessian matrix, at a scale that is automatically selected for each pixel. We discuss efficient implementation using a Fast Explicit Diffusion scheme and demonstrate results on synthetic images and vessels in fundus images. Compared to previous isotropic and anisotropic fourth-order filters, as well as established second-order vessel enhancing filters, our newly proposed one better restores the centerlines in all cases.

Long-time behavior of a finite volume discretization for a fourth order diffusion equation

We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker–Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.

Unconditional stability of alternating difference schemes with intrinsic parallelism for two-dimensional fourth-order diffusion equation

In this paper, the parallel difference schemes for parabolic equation are studied. The general alternating difference schemes with intrinsic parallelism for two-dimensional fourth-order diffusion equation are presented and proved to be unconditionally stable by the energy method. Two kinds of alternating band explicit (ABE1, ABE2) schemes, alternating band explicit–implicit (ABE–I) scheme and alternating band Crank–Nicolson (ABC–N) scheme are the special cases of the general schemes constructed here. Numerical experiments are performed to examine the accuracy and unconditional stability.

Fluid ordering effects and density variations in nanochannel flows: a quasicontinuum theory

A quasicontinuum self‐diffusion theory that can capture the fluid ordering effects and density variations predicted by nonequilibrium molecular dynamics in nanochannel flows is presented. The physics of the problem suggests a fourth order diffusion equation for the concentration as a function of position and time requiring the classic diffusion coefficient D and a microstructural internal length g that relates directly to the shape of the molecular potential of the molecular dynamics simulations. Given the appropriate initial and boundary conditions, the aforesaid linear partial differential equation is analytically solved for the one‐dimensional steady state, whereas the case of symmetric as well as asymmetric wall conditions has been accounted for by examining different inhomogeneous diffusion distributions. Copyright © 2013 John Wiley & Sons, Ltd.

Fourth Order Diffusion Equations with Increasing Entropy

The general quasi-linear autonomous fourth order diffusion equation ut = −[G(u)uxxx + h(u, ux, uxx)]x with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may have any positive twice-differentiable diffusivity function G(u). The forms of such equations are the indicators of more general conservation equations whose entropy equation may be expressed in an alternative reaction-diffusion form whose source term, although reducible, is positive.

Stability, dynamics, and the onset of multi-pulsing in a mode-locked laser cavity using phase-sensitive amplification

An averaged evolution equation is presented and its dynamics studied for a mode-locked laser where the intensity discrimination (saturable absorption) in the cavity is provided by phase-sensitive amplification. The phase-sensitive amplifier acts as a phase-filter for selecting the specific intensity-dependent phase-rotation of the mode-locked pulse that locks the phase to the amplifier pump phase. The resulting averaged equation is a Swift–Hohenberg type model which is a fourth-order diffusion equation with cubic-quintic nonlinearities. Additionally, the governing evolution has a new linear growth term which couples to the nonlocal cavity energy. This parameter is a standard bifurcation parameter in Swift–Hohenberg models and is controlled by the cavity saturable gain. Such a modification to the governing evolution is the first of its kind to be considered theoretically in the context of the Swift–Hohenberg equation, and its significant impact on the mode-locked pulse dynamics and multi-pulsing behavior is explored.

Cahn–Hilliard Inpainting and a Generalization for Grayvalue Images

The Cahn–Hilliard equation is a nonlinear fourth order diffusion equation originating in material science for modeling phase separation and phase coarsening in binary alloys. The inpainting of binary images using the Cahn–Hilliard equation is a new approach in image processing. In this paper we discuss the stationary state of the proposed model and introduce a generalization for grayvalue images of bounded variation. This is realized by using subgradients of the total variation functional within the flow, which leads to structure inpainting with smooth curvature of level sets.

Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form

The entropy evolution behaviour of a partial differential equation (PDE) in conservation form, may be readily discerned from the sign of the local source term of Shannon information density. This can be easily used as a diagnostic tool to predict smoothing and non-smoothing properties, as well as positivity of solutions with conserved mass. The familiar fourth order diffusion equations arising in applications do not have increasing Shannon entropy. However, we obtain a new class of nonlinear fourth order diffusion equations that do indeed have this property. These equations also exhibit smoothing properties and they maintain positivity. The counter-intuitive behaviour of fourth order diffusion, observed to occur or not occur on an apparently ad hoc basis, can be predicted from an easily calculated entropy production rate. This is uniquely defined only after a technical definition of the irreducible source term of a reaction diffusion equation.

Non-homogeneous boundary conditions for a fourth-order diffusion equation

The existence of classical solutions to a one-dimensional non-linear fourth-order elliptic equation arising in quantum semiconductor modeling is proved for a class of non-homogeneous boundary conditions using degree theory. Furthermore, some non-existence results for other classes of boundary conditions are presented. To cite this article: P. Amster et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Computational Modeling of Mineral Unmixing and Growth

A new finite element based simulation technique for mineral growth governed by the classical Cahn–Hilliard equation is presented. The particular format of the underlying Flory–Huggins free energy for non-ideal mixtures is characterized through a double-well potential. It allows for uphill diffusion driven by gradients in the chemical potential and thus provides the appropriate framework to simulate phase separation typically encountered in mineral unmixing and growth. For the finite element discretization, the governing fourth order diffusion equation is reformulated in terms of a system of two coupled second order equations. For the temporal discretization, a heuristic adaptive time stepping scheme is applied in order to simulate not only the early stages of phase separation but also the long term behavior of ageing and grain fusion. The basic features of the Cahn–Hilliard equation are elaborated by means of selected geologically relevant examples. In particular, isotropic and anisotropic mineral growth and symplectite formation are studied and the long term response in the sense of Ostwald ripening is illustrated.