Nonlinear Reaction-diffusion Problems Research Articles

A nodally bound-preserving finite element method

Abstract This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.

Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions

<p style='text-indent:20px;'>The main goal of this paper is to introduce and analyze a new nonlocal reaction-diffusion model with in-homogeneous Neumann boundary conditions. We prove the existence and uniqueness of a solution in the class <inline-formula><tex-math id="M1">\begin{document}$ C((0, T], L^\infty(\Omega)) $\end{document}</tex-math></inline-formula> and the dependence on the data. Proofs are based on the Banach fixed-point theorem. Our results extend the results already proven by other authors. A numerical approximating scheme and a series of numerical experiments are also presented in order to illustrate the effectiveness of the theoretical result. The overall scheme is explicit in time and does not need iterative steps; therefore it is fast.</p>

Monotone iterative technique for solving finite difference systems of time fractional parabolic equations with initial/periodic conditions

This article is devoted to a numerical analysis of two types of fractional nonlinear reaction-diffusion problems with periodic conditions or with initial conditions. We also consider two types of derivative in time, the first one is that of Caputo and the second is the conformable derivative; we discretize the problem via a finite difference method. We construct two iterative schemes by the upper and lower solutions method which converges monotonically towards a maximal solution or a minimal solution of the considered problem when the mesh decreases to zero, depending on whether the initial iteration is an upper solution or a lower solution. A comparison lemma for the different monotone sequences is also proved. The presented iterative scheme is used to show that the finite difference system converges to continuous solutions of the fractional reaction-diffusion problem. Finally to validate the theoretical results some examples with numerical simulations of reaction-diffusion problem are also presented and discussed in detail.

Automated computation of topological derivatives with application to nonlinear elasticity and reaction–diffusion problems

While topological derivatives have proven useful in applications of topology optimization and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. We present a systematic yet formal approach for the numerical computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction–diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behavior also for other, previously unknown problems, indicating the correctness of the procedure.

A posteriori error estimates of goal-oriented adaptive finite element methods for nonlinear reaction–diffusion problems

In this paper, we construct two goal-oriented a posteriori error estimates with finite element approximations in the enriched space for the reaction–diffusion equation with the nonlinear reaction term. The first error estimator is proved to be the rigorous global lower and upper bounds on the error in the quantity of interest, and the second one is computed simply and efficiently. We propose the corresponding goal-oriented adaptive finite element algorithms, and show the effectiveness and the similar performance of those methods with a series of numerical experiments.

Finite Element Method and numerical study for a super-linear reaction-diffusion problem with integral conditions

In this work, we prove the existence, uniqueness, and continuous dependence of generalized solution of a nonlinear reaction-diffusion problem with only integral terms in the boundaries, by using the finite element method.Also we have developed an efficient numerical finite difference schemes. Some numerical results are reported to show the efficiency and accuracy of the scheme.

Nonlinear nonlocal reaction-diffusion problem with local reaction

<p style='text-indent:20px;'>In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.</p>

Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

A preservative splitting approximation of the solution of a variable coefficient quenching problem

This paper studies the numerical solution of a two-dimensional quenching type nonlinear reaction-diffusion problem via dimensional splitting. The variable coefficient differential equation considered possesses a nonlinear forcing term, and may lead to strong quenching singularities that have profound multiphysics and engineering applications to the energy industry. Our investigations focus on the construction and preservative properties of a semi-adaptive Peaceman-Rachford procedure for solving the aforementioned problem. In the study, the multidimensional differential equation is decomposed into single dimensional subequations so that the computational efficiency can be effectively raised. Temporal adaptation is implemented through arc-length estimations of the rate-of-change of the numerical solution. The positivity, monotonicity and localized linear stability of the variable step splitting scheme are analyzed. These ensure key preservations of underlying physical features of the computed approximations for applications. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and accuracy of the splitting method for solving singular quenching-combustion problems.

Morley FEM for the fourth-order nonlinear reaction-diffusion problems

Nonconforming Morley finite element method is applied to a fourth order nonlinear reaction-diffusion problems. After deriving some regularity results to be used subsequently in our error analysis, Morley FEM is employed to discretize in the spatial direction to obtain a semidiscrete problem. A priori bounds for the discrete solution are derived and with the help of an auxiliary problem, optimal error estimates are proved for the semidiscrete scheme. Based on backward Euler method, a completely discrete scheme is analysed and wellposedness of the discrete problem is discussed. A priori error bounds are derived. It is observed that constants do not depend exponentially on the inverse of a small parameter appeared as the coefficient in the fourth order term and all the results are derived when the initial data are in H02. Finally, some computational experiments are conducted to confirm our theoretical findings.

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

A hyper-reduction method using adaptivity to cut the assembly costs of reduced order models

At every iteration or timestep of the online phase of some reduced-order modelling schemes for non-linear or time-dependent systems, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The projected small linear systems are cheap to solve, but assembly and projection become the dominant computational cost. In this paper we introduce a new hyper-reduction strategy called reduced assembly (RA) that drastically cuts these costs. RA consists of a triangulation adaptation algorithm that uses a local error indicator to construct a reduced assembly triangulation specially suited to the reduced order basis. Crucially, this reduced assembly triangulation has fewer cells than the original one, resulting in lower assembly and projection costs. We demonstrate the efficacy of RA on a Galerkin-POD type reduced order model (RAPOD). We show performance increases up to five times over the baseline Galerkin-POD method on a non-linear reaction–diffusion problem solved with a semi-implicit time-stepping scheme and up to seven times for a 3D hyperelasticity problem solved with a continuation Newton–Raphson algorithm. The examples are implemented in the DOLFIN finite element solver using PETSc and SLEPc for linear algebra. Full code and data files to produce the results in this paper are provided as supplementary material.

Solving nonlinear reaction–diffusion problem in electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme systems using Taylor series method

A mathematical model of electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme is discussed. The model involves the coupled system of non-linear reaction–diffusion equations of substrate and hydrogen ion. The non-linear term in this model is related to the Michaelis–Menten reaction of the substrate and non-Michaelis–Menten kinetics of hydrogen ion. The approximate analytical expression of concentration of substrate and hydrogen ion has been derived by solving the non-linear reactions using Taylor’s series method. Reaction rate and effectiveness factor are also reported. A comparison between the analytical approximation and numerical solution is also presented. The effects of external mass transfer coefficient and the electrostatic potential on the overall reaction rate were also discussed.

A numerical algorithm based on modified orthogonal linear spline for solving a coupled nonlinear inverse reaction-diffusion problem

In this paper, a modified orthogonal linear spline (OL-spline) is used for the numerical solution of a coupled nonlinear inverse reaction-diffusion problem to determine the unknown boundary conditions. The convergence properties of the new linear combination are obtained. A quasi-linearization technique is utilized to linearize the nonlinear term in the equations. This process produces a linear system of equations which can be solved easily. Using the new inequalities, error estimation and convergence of the proposed method are investigated. Two numerical examples are given to demonstrate the computational efficiency of the method and also the experimental convergence rate of examples are obtained.

A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics

<p style='text-indent:20px;'>This work gives a posteriori error estimates for a finite volume implicit scheme, applied to a two-time nonlinear reaction-diffusion problem in population dynamics, whose evolution processes occur at two different time scales, represented by a parameter <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula> small enough. This work consists of building error indicators concerning time and space approximations and using them as a tool of adaptive mesh refinement in order to find approximate solutions to such models, in population dynamics, that are often hard to be handled analytically and also to be approximated numerically using the classical approach. <p style='text-indent:20px;'>An application of the theoretical results is provided to emphasize the efficiency of our approach compared to the classical one for a spatial inter-specific model with constant diffusivity and population growth given by a logistic law in population dynamics.

The Uniqueness of Stationary Solution for Nonlinear Random Reaction-diffusion Equation

We study a nonlinear random reaction-diffusion problem in abstract Banach spaces, driven by a real noise, with random diffusion coefficient and random initial condition. We consider a polynomial non linear term. The reaction-diffusion equation belongs to the class of parabolic stochastic partial differential equations. We assume that the initial condition is an element of Hilbert space. The real noise is a Wiener process. We construct a suitable stochastic basis and define the solution of reaction-diffusion problem in the weak sense. We define the stationary process in abstract Banach spaces in the strong sense of Doob-Rozanov. That is, the probability density function of the stochastic process is independent of time shift. We define the invariant measure for random reaction-diffusion equation in the sense of Arnold, DaPrato, and Zabczyk [1,2]. In other words, we define the invariant measure for random dynamical system, associated with random reaction-diffusion problem.&#x0D; Using the Variation Inequalities Theory, we prove the uniqueness of stationary solution for nonlinear random reaction-diffusion problem. The obtained theoretical results have several applications in Quantum Physics, Biology, Medicine, and Economic Sciences. Especially, we can study the existence of stationary solution for the stochastic models of tumor growth.

PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the method

To extend prevailing scaling limits when solving time-dependent partial differential equations, the parallel full approximation scheme in space and time (PFASST) has been shown to be a promising parallel-in-time integrator. Similar to space–time multigrid, PFASST is able to compute multiple time-steps simultaneously and is therefore in particular suitable for large-scale applications on high performance computing systems. In this work we couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator. While PFASST provides global, large-scale “parallelization across the step”, the inner parallel SDC method allows integrating each individual time-step “parallel across the method” using a diagonalized local Quasi-Newton solver. This new method, which we call “PFASST with Enhanced concuRrency” (PFASST-ER), therefore exposes even more temporal concurrency. For two challenging nonlinear reaction-diffusion problems, we show that PFASST-ER works more efficiently than the classical variants of PFASST and can use more processors than time-steps.

Numerical resolution of a pseudo-parabolic Buckley-Leverett model with gravity and dynamic capillary pressure in heterogeneous porous media

We discuss a numerical approach for the simulation of a hydrodynamic model for immiscible incompressible two-phase flow in heterogeneous porous media to deal with effects of dynamic capillary pressure and gravity. The governing system of equations consists of a nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem. Upon certain manipulation of the equations, we rewrite the transport model as a nonlinear elliptic reaction-diffusion problem along with a simpler time-dependent relation to the dynamic capillary pressure. Hybridized mixed finite elements and domain decomposition procedures are used for the spatial discretization of the equations. Proper locally conservative finite volume techniques are used for a balancing discretization of the first-order hyperbolic flux and the dispersive behavior inherent of the pseudo-parabolic model. After discretizing in time with an implicit method, the resulting nonlinear algebraic equations is solved with a sequential time-marching approach and the arising systems of linear equations are solved efficiently with an algebraic multigrid method. Thus, we formulate and test numerically a fully coupled and implicit formulation of the transport problem of pseudo-parabolic nature. We find that the new procedure is accurate, robust, efficient and easy for implementation. We simulated the two-dimensional transport of the full nonlinear modified Buckley-Leverett model with a dynamic capillary pressure model in a variety of flow regimes along with heterogeneous permeability fields and domination of gravity. Our numerical experiments demonstrated the viability of the proposed formulation based on computational simulations of typical flow situations and saturation overshoot.

A polynomial differential quadrature-based numerical scheme to simulate the nerve pulse propagation in the spatial Fitzhugh-Nagumo model

Nonlinear dynamics connect the neurons that form the brain, and thus, the complex information is produced and transported. The function of the neurons and the problem of understanding the dynamics of the brain has been the research area of mathematical neuroscience. In this study, the modelling and simulation of the propagation of the electric field based Action Potential (AP) on the Two Dimensional (2-D) field of axon network, whose matrix consists of 128×128 electrically coupled neurons were done using nonlinear Spatial FitzHugh Nagumo (SFN) equations. SFN equations are a particular class of Partial Differential Equation’s (PDE’s) exhibiting travelling wave behaviour occurred in neuron systems. The motivation of this paper is to evaluate the SFN equation, which is a special kind of the time-dependent nonlinear reaction-diffusion problem governing neuron dynamics numerically in 2-D space addressed by investigating the Polynomial-based Differential Quadrature Method (PDQM) having Chebyshev-Gauss-Lobatto quadrature points. The solution occurs as elliptical spiral waves induced by electrical stimulation. Thus, the neuronal system behaviour and the interaction with the specific type of Boundary Conditions (BC’s) are predicted. The space derivatives are discretised through PDQM. In this way, the problem is reduced into a system of first-order non-linear differential equations. Hereafter the time derivatives of the SFN equation are solved through the Finite Difference Method (FDM). The various dynamical behaviour that governs the travelling wave pattern regarding the Initial Condition’s (IC’s), BC’s and way of stimulation of the neuron model examined in details. Numerical results indicated that the proposed PDQM provide reliable, fast, and efficient solutions.