Parabolic Differential Equations Research Articles

Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

Characterizations of parabolic Muckenhoupt classes

This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of p-Laplace type. The main results include characterizations for the limiting parabolic A∞ and A1 classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calderón–Zygmund decompositions, covering and chaining arguments appear in the proofs.

Space-Time Least-Squares Finite Element Methods for Parabolic Distributed Optimal Control Problems

Abstract We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier. Our final formulation fulfills the Babuška–Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem. Numerical experiments confirm our theoretical findings.

Explicit stabilized multirate methods for the monodomain model in cardiac electrophysiology

Fully explicit stabilized multirate (mRKC) methods are well-suited for the numerical solution of large multiscale systems of stiff ordinary differential equations thanks to their improved stability properties. To demonstrate their efficiency for the numerical solution of stiff, multiscale, nonlinear parabolic PDE's, we apply mRKC methods to the monodomain equation from cardiac electrophysiology. In doing so, we propose an improved version, specifically tailored to the monodomain model, which leads to the explicit exponential multirate stabilized (emRKC) method. Several numerical experiments are conducted to evaluate the efficiency of both mRKC and emRKC, while taking into account different finite element meshes (structured and unstructured) and realistic ionic models. The new emRKC method typically outperforms a standard implicit-explicit baseline method for cardiac electrophysiology. Code profiling and strong scalability results further demonstrate that emRKC is faster and inherently parallel without sacrificing accuracy.

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

Implicit–explicit (IMEX) Euler formula together with finite element methods is proposed to solve numerically rather general semilinear parabolic differential equations with initial data in Banach space Xα, 0<α≤1. The time-space regularity of solutions to this class of equations is first investigated. Stability and error estimates are provided for this fully discrete scheme by discrete semigroup method. In the special case when X is the Hilbert space L2(Ω), these error estimates bridge the gap between the existing results on this scheme for semilinear parabolic problems with initial data in L2(Ω) and in H1(Ω) under general nonlinearity. Numerical experiments for nonlinear Allen–Cahn equations verify and complement our theoretical results.

Stabilization of coupled delayed nonlinear time fractional reaction diffusion systems using sampled‐in‐space sensing and actuation

AbstractThis paper is considered with the asymptotic stabilization of coupled delayed nonlinear time fractional reaction diffusion systems (FRDSs) governed by fractional parabolic partial differential equations (PDEs) with space‐dependent coefficients under sampled‐data in space control. It is assumed that state measurements can be averaged measurements (AMs) or point measurements (PMs), and a finite number of sensing and actuation devices are located in a spaced manner along the spatial domain of the interest. With the proposed sampled‐data in space controller, the closed‐loop stability is obtained. Tuning rules of system parameters and control parameters are derived using the fractional Halanay's inequality and the fractional Lyapunov method. Subsequently, the dual problem of observer design is formulated. Fractional examples are used to valid the theoretical result. Discussions on the extension of sampled‐data boundary feedback stabilization are provided finally.

Structure-preserving reduced order model for cross-diffusion systems

In this work, we construct a structure-preserving Galerkin reduced-order model for the resolution of parametric cross-diffusion systems. Cross-diffusion systems are often used to model the evolution of the concentrations or volumic fractions of mixtures composed of different species, and can also be used in population dynamics (as for instance in the SKT system). These systems often read as nonlinear degenerated parabolic partial differential equations, the numerical resolutions of which are highly ex- pensive from a computational point of view. We are interested here in cross-diffusion systems which exhibit a so-called entropic structure, in the sense that they can be formally written as gradient flows of a certain entropy functional which is actually a Lyapunov functional of the system. In this work, we propose a new reduced-order modelling method, based on a reduced basis paradigm, for the resolution of parameter-dependent cross-diffusion systems. Our method preserves, at the level of the reduced-order model, the main mathematical properties of the continuous solution, namely mass conservation, non- negativeness, preservation of the volume-filling property and entropy-entropy dissipation relationship. The theoretical advantages of our approach are illustrated by several numerical experiments.

A model‐based failure times identification for a system governed by a 2D parabolic partial differential equation

AbstractThis research focuses on the identification of failure times in thermal systems governed by partial differential equations, a task known for its complexity. A new model‐based diagnostic approach is presented that aims to accurately identify failing heat sources and accurately determine their failure times, which is crucial when multiple heat sources fail and there is a delay in detection by distant sensors. To validate the effectiveness of the approach, a comparative analysis is carried out with an established method based on a Bayesian filter, the Kalman filter. The aim is to provide a comprehensive analysis, highlighting the advantages and potential limitations of the methodology. In addition, a Monte Carlo simulation is implemented to assess the impact of sensor measurements on the performance of this new approach.

An alternative approach to solenoidal Lipschitz truncation

In this work, we present an alternative approach to obtain a solenoidal Lipschitz truncation result in the spirit of D. Breit, L. Diening and M. Fuchs [Solenoidal Lipschitz truncation and applications in fluid mechanics. J. Differ. Equ. 253 (2012), 1910–1942.]. More precisely, the goal of the truncation is to modify a function $u \in W^{1,p}(\mathbb {R}^N;\mathbb {R}^N)$ that satisfies the additional constraint $\operatorname {div} u=0$ , such that its modification $\tilde {u}$ is Lipschitz continuous and divergence-free. This approach is different to the approaches outlined in the aforementioned work and D. Breit, L. Diening and S. Schwarzacher [Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23 (2013), 2671–2700, Section 4] and is able to obtain the rather strong bound on the difference between $u$ and $\tilde {u}$ from the former article. Finally, we outline how the approach pursued in this work may be generalized to closed differential forms.

Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint

Abstract We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d \mathbb{R}^{d} , d ≥ 1 d\geq 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

This paper presents a study on the numerical solutions of the Burgers’ equation with forcing effects. The article proposes three hybrid methods that combine two-point, three-point, and four-point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers’ equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers’ problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers’ equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers-type parabolic partial differential equations arising in various fields of science and engineering.

Sharp analysis of [formula omitted] method on graded mesh for time fractional parabolic differential equation

In this paper, we study the L1−2 method for solving numerically a class of time fractional parabolic differential equations on a graded mesh. Based on the nonsmooth regularity assumptions, the stability and convergence of the proposed numerical method are investigated. Using rigorous and sharp theoretical analysis, we obtain the optimal graded parameter r=(3−α)/(1+α) of the graded mesh. Numerical experiments are given to demonstrate that even if the α is small, the graded mesh based on the optimal graded parameter is still able to recover the optimal convergence order of the L1−2 method.

Discontinuous Galerkin Method of the First Order Parabolic Differential Equation

This paper is concerned with the mathematical study of the discontinuous Galerkin (DG) finite element method for the parabolic differential equation. The DG method is a vital numerical method with much mass compensation and more flexible meshing than other methods. In this study, we give a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The error analysis of this problem is also established. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay

ABSTRACT In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems.

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

Numerical investigation of high-dimensional option pricing PDEs by utilizing a hybrid radial basis function - finite difference procedure

The target of this research is to resolve high-dimensional partial differential equations (PDEs) for multi-asset options, modeled as parabolic time-dependent PDEs. We present a hybrid radial basis function - finite difference (RBF-FD) solver, which combines the advantages of Gaussian and multiquadric functions. Additionally, we employ the Krylov subspace method on the resulting system of ordinary differential equations, reducing the computation load for finding the numerical solution. Computational tests, up to seven dimensions, and comparisons support the superiority of our hybrid solver.

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition - that we decide to call here local-time Kirchhoff's condition - induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

CUQIpy: II. Computational uncertainty quantification for PDE-based inverse problems in Python

Inverse problems, particularly those governed by Partial Differential Equations (PDEs), are prevalent in various scientific and engineering applications, and uncertainty quantification (UQ) of solutions to these problems is essential for informed decision-making. This second part of a two-paper series builds upon the foundation set by the first part, which introduced CUQIpy, a Python software package for computational UQ in inverse problems using a Bayesian framework. In this paper, we extend CUQIpy’s capabilities to solve PDE-based Bayesian inverse problems through a general framework that allows the integration of PDEs in CUQIpy, whether expressed natively or using third-party libraries such as FEniCS. CUQIpy offers concise syntax that closely matches mathematical expressions, streamlining the modeling process and enhancing the user experience. The versatility and applicability of CUQIpy to PDE-based Bayesian inverse problems are demonstrated on examples covering parabolic, elliptic and hyperbolic PDEs. This includes problems involving the heat and Poisson equations and application case studies in electrical impedance tomography and photo-acoustic tomography, showcasing the software’s efficiency, consistency, and intuitive interface. This comprehensive approach to UQ in PDE-based inverse problems provides accessibility for non-experts and advanced features for experts.

Numerical Analysis for a Singularly Perturbed Parabolic Differential Equation with a Time Delay

Numerical Analysis for a Singularly Perturbed Parabolic Differential Equation with a Time Delay

Stability and Error Analysis of an Efficient Numerical Method for Convection Dominated Parabolic PDEs with Jump Discontinuity in Source Function on Modified Layer-Adapted Mesh

Stability and Error Analysis of an Efficient Numerical Method for Convection Dominated Parabolic PDEs with Jump Discontinuity in Source Function on Modified Layer-Adapted Mesh