Non-smooth Solutions Research Articles

Eloquent numerical approach for solving generalized time fractional convection-diffusion-reaction problems

Abstract This paper attempts to develop a spectral method based on derivatives of orthogonal polynomials to solve the time fractional convection–diffusion-reaction equations. The method utilizes derivatives of fractional order orthogonal functions to approximate derivatives involved in the fractional differential equations. Specifically, the derivatives of fractional order Legendre functions and fractional order Chebyshev functions are used to represent both integer and non-integer derivatives of the solutions. These derivative representations are achieved through the use of operational matrices, which are matrices that encode the operations performed on the polynomials. An attempt is made to derive the operational matrix of Vieta-Fibonacci-like polynomials and used to solve the fractional differential equations. The derived operational matrix provides a systematic way to manipulate and work with these polynomials, facilitating their application in various mathematical and engineering problems. By employing these operational matrices, the original fractional convection–diffusion-reaction equation is transformed into a system of linear or nonlinear algebraic equations. However, if the system is nonlinear, a Newton-like solver is applied, which is capable of handling nonlinear systems. The estimation of error bounds of numerical solutions is also given. The numerical experiments have been performed over a few test examples to validate the proposed numerical method. The use of fractional order functions highlights their ability to solve fractional differential equations with non-smooth solutions accurately.

A New Operational Matrices Based-Technique for Fractional Integro Reaction-Diffusion Equation Involving Spatiotemporal Variable-Order Derivative

In this article, a novel generalized model of the nonlinear fractional integro-reaction-diffusion equation (NFIRDE) with spatiotemporal variable-order (SVO) is introduced, where the variable order derivatives are equipped with the Atangana–Baleanu–Caputo (ABC) sense. This model represents a great generalization of a significant type of NFIRDE and their applications. Moreover, A novel, efficient and fully spectral shifted Legendre tau technique is developed to solve the proposed model. Despite the difficulty of applying this mechanism to solve this type of equations, due to the presence of nonlinear terms and the SVO functions that appear in the traditional differential and integral operational matrices. We deduce some new operational matrices that play the fundamental role in facilitating the implementation of the tau method. These operational matrices represent the SVO ABC-derivative, the integro term within the model, as well as the vector multiplications with the space-time Kronecker product. As a result, the proposed model is restructured into a system of nonlinear algebraic equations, which simplifying the solving process. We illustrate our method’s effectiveness and validity with numerical examples with both smooth and non-smooth solutions. Our findings show that the proposed tau method delivers accurate results and exhibits non-local properties.

An operational discontinuous Galerkin shallow water model for coastal flood assessment

Hydrodynamic modeling for coastal flooding risk assessment is a highly relevant topic. Many operational tools available for this purpose use numerical techniques and implementation paradigms that reach their limits when confronted with modern requirements in terms of resolution and performances. In this work, we present a novel operational tool for coastal hazards predictions, currently employed by the BRGM agency (the French Geological Survey) to carry out its flooding hazard exposure studies and coastal risk prevention plans on International and French territories. The model, called UHAINA (wave in the Basque language), is based on an arbitrary high-order discontinuous Galerkin discretization of the nonlinear shallow water equations with SSP Runge–Kutta time stepping on unstructured triangular grids. It is built upon the finite element library AeroSol, which provides a modern C++ software architecture and high scalability, making it suitable for HPC applications. The paper provides a detailed development of the mathematical and numerical framework of the model, focusing on two key-ingredients : (i) a pragmatic P0 treatment of the solution in partially dry cells which garantees efficiently well-balancedness, positivity and mass conservation at any polynomial order; (ii) an artificial viscosity method based on the physical dissipation of the system of equations providing nonlinear stability for non-smooth solutions. A set of numerical validations on accademic benchmarks is performed to highlight the efficiency of these approaches. Finally, UHAINA is applied on a real operational case of study, demonstrating very satisfactory results.

Analysis of acoustic radiation problems involving arbitrary immersed media interfaces by the extended finite element method with Dirichlet to Neumann boundary condition

To quantify the influence of moving immersed media on acoustic radiation, this study develops an efficient method for acoustic radiation with arbitrary immersed media interfaces based on the extended finite element method (XFEM) and the Dirichlet-to-Neumann (DtN) boundary condition. The XFEM is employed for efficient and accurate modeling of the acoustic field with boundary shape variations. It requires no modification of the computational mesh and accurately captures non-smooth solutions on the interface by constructing enrichment functions. Additionally, the DtN boundary condition simulates the far-field radiation condition by establishing the relationship between the acoustic pressure and its derivatives. Numerical examples show that the proposed method efficiently characterizes changes in the position of immersed media interfaces without re-meshing the mesh. Variations in the thickness of porous material domains alter the acoustic radiation characteristics, with thicker porous material domains resulting in more pronounced noise reduction effects. Compared to changes in the thickness of porous material domains, changes in their position significantly alter the distribution of radiation pressure, indicating that ideal noise reduction effects can be achieved by strategically placing porous materials in specific locations in practical engineering applications.

Superconvergent method for weakly singular Fredholm-Hammerstein integral equations with non-smooth solutions and its application

In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results.

High-Order Approximation to Caputo Derivative on Graded Mesh and Time-Fractional Diffusion Equation for Nonsmooth Solutions

Abstract In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations (TFDEs) involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the nonuniform mesh. The truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as min{4−α,rα} and (4−α)/α, respectively, where α∈(0,1) is the order of fractional derivative and r≥1 is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on the graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general r. Then, we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for r = 1, is analyzed for nonsmooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.

Deep finite volume method for partial differential equations

In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order ≥2) partial differential equations (PDEs). Our approach centers on a novel loss function crafted from local conservation laws derived from the original PDE, distinguishing DFVM from traditional deep learning methods. By formulating DFVM in the weak form of the PDE rather than the strong form, we enhance accuracy, particularly beneficial for PDEs with less smooth solutions compared to strong-form-based methods like Physics-Informed Neural Networks (PINNs). A key technique of DFVM lies in its transformation of all second-order or higher derivatives of neural networks into first-order derivatives which can be computed directly using Automatic Differentiation (AD). This adaptation significantly reduces computational overhead, particularly advantageous for solving high-dimensional PDEs. Numerical experiments demonstrate that DFVM achieves equal or superior solution accuracy compared to existing deep learning methods such as PINN, Deep Ritz Method (DRM), and Weak Adversarial Networks (WAN), while drastically reducing computational costs. Notably, for PDEs with nonsmooth solutions, DFVM yields approximate solutions with relative errors up to two orders of magnitude lower than those obtained by PINN. The implementation of DFVM is available on GitHub at https://github.com/Sysuzqs/DFVM.

Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations

The main objective of this study is to introduce an improvement of Picard’s method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo’s fractional derivative. Using the Picard’s method to solve fractional differential equations is straightforward. However, dealing with the integral in each Picard’s iteration becomes tough or even impossible for nonlinear problems. Thus, we propose an iterative strategy called the fractional Jacobi–Picard iteration method, which combines Picard’s iteration method with the shifted Jacobi polynomial. The computation of the fractional integrals of the shifted Jacobi polynomials is easily achieved at each step by utilizing properties of the fractional integral and shifted Jacobi polynomial. Furthermore, this approach not only transforms the system of equations into a reversible form but also solves it using the Gauss–Seidel technique. The convergence analysis of the method has been carefully performed. We performed detailed numerical simulations to show how well our method performs compared to other methods. Our results demonstrate the effectiveness and accuracy of our approach, especially in handling problems with non-smooth solutions.

Second-order error analysis of a corrected average finite difference scheme for time-fractional Cable equations with nonsmooth solutions

In this paper, we first formulate an average finite difference scheme with appropriate correction terms for the time-fractional initial-value problem of y′(t)+0RLDt1−αy(t)=g(t), where 0RLDt1−α denotes the Riemann–Liouville fractional derivative of order 1−α(α∈(0,1)). It is shown that, under some suitable conditions on the data, the numerical solution converges to the exact solution with order O(τ2), where τ is the time step size. The method is then extended to solve the time-fractional Cable equation, combined with a standard discretisation of the spatial derivatives on a uniform mesh. The second-order time convergence rate is proved. Several numerical examples are given to illustrate the good agreement with the theoretical analysis of the presented methods.

A meshless approach based on fractional interpolation theory and improved neural network bases for solving non-smooth solution of 2D fractional reaction–diffusion equation with distributed order

The primary objective of this research paper is to present a novel and effective meshless numerical approach for solving the 2D time fractional reaction diffusion system with distributed order on an arbitrary domain. Gauss–Legendre quadrature formula is applied to discretize distributed-order derivative integral. We establish the piecewise parabolic fractional interpolation theory and with its assistance, the proposed approach can proficiently solve the non-smooth solutions of the equations and more accurately approximate the Caputo fractional derivative. This meshless method based on improved neural network bases combines the high accuracy approximation advantage and strong express ability of neural network to construct the basis functions set on arbitrary domains, which significantly reduces a computational consumption. The bases constructed based on the neural network allow the selection of 12 bases numbers to achieve an approximation equivalent to that of 1000 ordinary bases. The theoretical analyses of error and convergence order for the meshless approach are carried out. Numerical examples are implemented to validate the high precision and capability of the meshless numerical approach.

Beyond quantum annealing: optimal control solutions to maxcut problems

Quantum Annealing (QA) relies on mixing two Hamiltonian terms, a simple driver and a complex problem Hamiltonian, in a linear combination. The time-dependent schedule for this mixing is often taken to be linear in time: improving on this linear choice is known to be essential and has proven to be difficult. Here, we present different techniques for improving on the linear-schedule QA along two directions, conceptually distinct but leading to similar outcomes: 1) the first approach consists of constructing a Trotter-digitized QA (dQA) with schedules parameterized in terms of Fourier modes or Chebyshev polynomials, inspired by the Chopped Random Basis algorithm for optimal control in continuous time; 2) the second approach is technically a Quantum Approximate Optimization Algorithm (QAOA), whose solutions are found iteratively using linear interpolation or expansion in Fourier modes. Both approaches emphasize finding smooth optimal schedule parameters, ultimately leading to hybrid quantum–classical variational algorithms of the alternating Hamiltonian Ansatz type. We apply these techniques to MaxCut problems on weighted 3-regular graphs with N = 14 sites, focusing on hard instances that exhibit a small spectral gap, for which a standard linear-schedule QA performs poorly. We characterize the physics behind the optimal protocols for both the dQA and QAOA approaches, discovering shortcuts to adiabaticity-like dynamics. Furthermore, we study the transferability of such smooth solutions among hard instances of MaxCut at different circuit depths. Finally, we show that the smoothness pattern of these protocols obtained in a digital setting enables us to adapt them to continuous-time evolution, contrarily to generic non-smooth solutions. This procedure results in an optimized QA schedule that is implementable on analog devices.

Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions

Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.

Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions

In this paper, we delve into the regularity properties and the development of an efficient numerical strategy for addressing the nonlinear time-fractional Schrödinger equation. Initially, we embark on an examination of the solution’s regularity, followed by an investigation into regularity enhancement through the application of the Laplace and finite Fourier sine transforms. Subsequently, after the decomposition of u-u0, we introduce an innovative time two-grid fitted scheme designed for the equation’s resolution, which demonstrates unconditional stability and convergence in the L2 norm, achieving the global convergence order of O(τF2+τC4+h2). Here, τF and τC denote the temporal step sizes on the fine and coarse grids, respectively, while h represents the spatial step size. Notably, this is the inaugural application of such an approach for solving the nonlinear time fractional Schrödinger equation. Ultimately, our numerical experiments validate that this method not only retains computational efficiency but also significantly enhances convergence accuracy.

NEWTON’S METHOD IN CONSTRUCTING A SINGULAR SET OF A MINIMAX SOLUTION IN A CLASS OF BOUNDARY VALUE PROBLEMS FOR THE HAMILTON - JACOBI EQUATIONS

The non-smooth features of the minimax (generalized) solution of the considered class of Dirichlet problems for equations of Hamiltonian type are due to the existence of pseudo-vertices - singular points of the boundary of the boundary set. The paper develops analytical and numerical methods for constructing pseudo-vertices and their accompanying constructive elements, which include local diffeomorphisms generating pseudo-vertices, as well as markers - numerical characteristics of these points. For markers, an equation with a characteristic structure inherent in equations for fixed points is obtained. An iterative procedure based on Newton’s method for the numerical construction of its solution is proposed. The convergence of the procedure to the pseudovertex marker is proved. An example of the numerical-analytical construction of a minimax solution is given, illustrating the effectiveness of the developed approaches for constructing nonsmooth solutions of boundary value problems.

A Numerical Study of a Stabilized Hyperbolic Equation Inspired by Models for Bio-Polymerization

Abstract This report investigates a stabilization method for first order hyperbolic differential equations applied to DNA transcription modeling. It is known that the usual unstabilized finite element method contains spurious oscillations for nonsmooth solutions. To stabilize the finite element method the authors consider adding to the first order hyperbolic differential system a stabilization term in space and time filtering. Numerical analysis of the stabilized finite element algorithms and computations describing a few biological settings are studied herein.

Implicitly linear Jacobi spectral-collocation methods for two-dimensional weakly singular Volterra-Hammerstein integral equations

Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral-collocation method to solve two-dimensional weakly singular Volterra-Hammerstein integral equations based on smoothing transformation and implicitly linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the nonlinear Hammerstein term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. The weakly singular integral term is discretized by Jacobi Gauss quadrature formula which can absorb the weakly singular kernel function into the quadrature weight function and eliminate the influence of the weakly singular kernel on the method. Convergence analysis in the L∞-norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples.

FDM/FEM for nonlinear convection–diffusion–reaction equations with Neumann boundary conditions—Convergence analysis for smooth and nonsmooth solutions

This paper aims to present in a systematic form the stability and convergence analysis of a numerical method defined in nonuniform grids for nonlinear elliptic and parabolic convection–diffusion–reaction equations with Neumann boundary conditions. The method proposed can be seen simultaneously as a finite difference scheme and as a fully discrete piecewise linear finite element method. We establish second convergence order with respect to a discrete H1-norm which shows that the method is simultaneously supraconvergent and superconvergent. Numerical results to illustrate the theoretical results are included.

An easy-to-implement recursive fractional spectral-Galerkin method for multi-term weakly singular Volterra integral equations with non-smooth solutions

An easy-to-implement recursive fractional spectral-Galerkin method for multi-term weakly singular Volterra integral equations with non-smooth solutions

A space-time spectral approximation for solving two dimensional variable-order fractional convection-diffusion equations with nonsmooth solutions

The study focuses on the numerical solutions of two-dimensional variable-order fractional convection-diffusion equations, which combine the principles of diffusion and convection to describe the movement of particles, energy, other physical quantities within a system. The numerical solution is obtained using shifted Legendre Gauss–Lobatto and shifted Chebyshev Gauss–Radau collocation techniques. The convection-diffusion equation is transformed into a system of algebraic equations utilizing shifted Chebyshev Gauss–Radau and shifted Legendre Gauss–Lobatto nodes. Additionally, numerical test examples are presented to demonstrate the method’s efficacy to handle nonsmooth solutions to the given problems.