Non-smooth Solutions Research Articles

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.

Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.

Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor’s series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

Improved rates for a space–time FOSLS of parabolic PDEs

We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components (u1,u2)=(u,-∇xu)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(u_1,\ extbf{u}_2)=(u,-\ abla _\ extbf{x} u)$$\\end{document}. The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-norms of ∇xu1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla _\ extbf{x} u_1$$\\end{document} and u2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{u}_2$$\\end{document}, the (graph) norm of U contains the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-norm of ∂tu1+divxu2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u_1 +{{\\,\ extrm{div}\\,}}_\ extbf{x} \ extbf{u}_2$$\\end{document}. When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of u2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{u}_2$$\\end{document}. In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of ∂tu1+divxu2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u_1 +{{\\,\ extrm{div}\\,}}_\ extbf{x} \ extbf{u}_2$$\\end{document}, i.e., of the forcing term f=(∂t-Δx)u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=(\\partial _t-\\Delta _x)u$$\\end{document}. Numerical results show significantly improved convergence rates.

Alternating direction implicit approach for the two-dimensional time fractional nonlinear Klein–Gordon and Sine–Gordon problems

The aim of this article is to establish a numerical scheme to achieve the theoretical accuracy near the weak singularity at t=0 in solving the two-dimensional time-fractional nonlinear mixed diffusion-wave equation (TFNMDWE). The governing problem involves diffusion term with time-fractional Caputo derivative (TFCD) of order ν1(0<ν1<1), and wave term with TFCD of order ν2(1<ν2<2). To handle the singularity at t=0, we use linearized L1 method to discretize both the TFCDs on nonuniform time meshes. By using the nonuniform L1 method to approximate TFCD and central difference operator for the space derivative approximation, the considered problem is converted to an equivalent system of equations. Then, we use the Alternating Direction Implicit (ADI) approach to develop a numerical scheme for solving the resulting system of coupled equations. Further, we prove stability analysis of the scheme. Numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFNMDWEs with smooth and non-smooth exact solutions to describe the accuracy of numerical scheme. The illustrated examples confirm that the scheme has second-order accuracy in space, and order of convergence (OC) in time direction is min(2−ν1,3−ν2,γν1,γ(ν2−1)), where γ is the mesh grading parameter used in construction of the nonuniform meshes. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

A space-time discontinuous Galerkin discretization for the linear transport equation

We consider a full-upwind DG approximation in space and time for the linear transport equation. Based on our results for linear symmetric Friedrichs systems we establish inf-sup stability and convergence in a mesh-dependent DG norm, and we construct an error indicator with respect to this norm. Numerical results of test problems with known solution demonstrate the efficiency of the a priori and a posteriori results as well for smooth and for non-smooth solutions. Then, we show that by introducing suitable degrees of freedom on the space-time element boundaries the corresponding hybrid formulation yields a reduction to a considerably smaller linear system.

Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity at t=0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order α(1<α<2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L1 method (single-step) and Nonuniform Crank-Nicolson L1−2 method (multi-step). The L1 method has order of convergence (OC) min⁡(3−α,γα), where γ is the mesh grading parameter used in construction of the nonuniform mesh, and L1−2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD at t=0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min⁡(3−α,γα) and second order convergence, respectively for all 1<α<2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

A deep learning method for multi-material diffusion problems based on physics-informed neural networks

Since the solutions of the multi-material diffusion problems are not smooth, the general physics-informed neural network (PINN) method does not work well for this problem. In this paper, we first give the interface continuity conditions which are necessarily added to the loss function as a loss term. Then, to adapt PINN for solving the multi-material diffusion problems with a single neural network, we propose a domain separation strategy. Furthermore, a normalization strategy for the loss terms is proposed to improve the prediction accuracy of the trained network. By combining the above techniques, we present the improved PINN methods called DS-PINN and nDS-PINN which are novel and advance the application of PINN for non-smooth solutions. Numerical experiments verify the robustness and accuracy of the new methods. Moreover, the new methods also perform well for the interface problems with jump conditions.

Three-Dimensional Numerical Transfer Model Using a Monotonized Z-Scheme

The aim of this work is to investigate a three-dimensional second-order monotone finite-difference scheme for the transport equation. The investigation is conducted for model three-dimensional transport equations of an incompressible medium. The properties of the three-dimensional extension of the Z-scheme with nonlinear correction are studied in this work. This study is an extension of the author’s previous works, where a nonlinear correction of one-dimensional transport equations was constructed. The proposed scheme uses “skew differences” containing values from different time layers instead of fluxes for the correction. The monotonicity of the obtained nonlinear scheme is numerically studied for a family of limiter functions for both smooth and non-smooth continuous solutions. The constructed scheme is absolutely stable but loses the monotonicity property when the Courant step is exceeded. The distinctive feature of the proposed finite-difference scheme is the minimalism of its template. The constructed numerical scheme is designed for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems that arise in solving such equations is the numerical modeling of strongly non-stationary medium- and small-scale processes in the low-latitude ionosphere of the Earth under conditions of the occurrence of the Rayleigh–Taylor instability and other types of instabilities, leading to the phenomenon of F‑scattering. Due to the fact that transport processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma is incompressible in the direction perpendicular to the magnetic field.

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

A Linearized L1-Galerkin FEM for Non-smooth Solutions of Kirchhoff Type Quasilinear Time-Fractional Integro-Differential Equation

A Linearized L1-Galerkin FEM for Non-smooth Solutions of Kirchhoff Type Quasilinear Time-Fractional Integro-Differential Equation

A robust higher-order numerical technique with graded and harmonic meshes for the time-fractional diffusion–advection–reaction equation

The solution of a time-fractional differential equation usually exhibits a weak singularity near the initial time t=0. It causes traditional numerical methods with uniform mesh to typically lose their accuracy. The technique of nonuniform mesh based on reasonable regularity of the solution has been found to be a very effective way to recover the accuracy. The current work aims to devise a compact finite difference scheme with temporal graded and harmonic meshes for solving time-fractional diffusion–advection–reaction equations with non-smooth solutions. The time-fractional operator involved in this is taken in the Caputo sense. A theoretical analysis of the stability and convergence of the proposed numerical technique is presented by von Neumann’s method. The scheme’s proficiency, robustness, and effectiveness are examined through three numerical experiments. A comparison of the numerical results on the uniform, graded, and harmonic meshes is presented to demonstrate the advantage of the suggested meshes over the uniform mesh. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.

A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions

In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence order O(N−2) in L∞ norm is proved, where N refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method.

Semi-conservative high order scheme with numerical entropy indicator for intrusive formulations of hyperbolic systems

This work considers high order discretizations for the intrusive stochastic Galerkin and polynomial moment method. Applications to hyperbolic systems result in solutions that typically involve a large number of wave interactions that must be resolved numerically. In order to reduce numerical oscillations, analytical and numerical entropy indicators are used to perform CWENO-type reconstructions in characteristic variables, when and where non-smooth solutions arise. The proposed method is analyzed for random isentropic Euler equations. In particular, a semi-conservative scheme is employed for non-polynomial pressure in order to reduce the computational cost, while still ensuring correct shock speeds.

Investigating and Mitigating Failure Modes in Physics-Informed Neural Networks (PINNs)

This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach is the requirement for manual hyperparameter tuning, making it impractical in the absence of validation data or prior knowledge of the solution. Our investigations of the loss landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate. Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence. To address these challenges, we introduce a novel method that bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients. Consequently, we reduce the dimension of the search space and make learning PDEs with non-smooth solutions feasible. Our method also provides a mechanism to focus on complex regions of the domain. Besides, we present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model's prediction, with adaptive and independent learning rates inspired by adaptive subgradient methods. We apply our approach to solve various linear and non-linear PDEs.

Local well-posedness and decay for some generalized shallow water equations

Consideration herein is a big class of nonlocal partial differential equations including some classical shallow water models such as b-family equation, Novikov equation, rotation-Camassa-Holm equation, generalized Camassa-Holm equation and Camassa-Holm-Novikov equation over a whole line. Based on the local structure of the dynamics along the characteristics, a local well-posedness result is established for some initial data class containing certain non-smooth solitary wave solutions. Subsequently in this set certain decay persistence properties are also constructed. These partially extend some obtained results and bring the effect of nonlinearities to more extent.