Time-dependent Partial Differential Equations Research Articles

Modeling and solution of eigenvalue problems of laminated cylindrical shells consisting of nanocomposite plies in thermal environments

AbstractThis work is dedicated to the modeling and solution of eigenvalue problems within shear deformation theory (SDT) of laminated cylindrical shells containing nanocomposite plies subjected to axial compressive load in thermal environments. In this study, the shear deformation theory for homogeneous laminated shells is extended to laminated shells consisting of functionally graded (FG) nanocomposite layers. The nanocomposite plies of laminated cylindrical shells (LCSs) are arranged in a piecewise FG distribution along the thickness direction. Temperature-dependent material properties of FG-nanocomposite plies are estimated through a micromechanical model, and CNT efficiency parameters are calibrated based on polymer material properties obtained from molecular dynamics simulations. After mathematical modeling, second-order time-dependent and fourth-order coordinate-dependent partial differential equations are derived within SDT, and a closed-form solution for the dimensionless frequency parameter and critical axial load is obtained for first time. After the accuracy of the applied methodology is confirmed by numerical comparisons, the unique influences of ply models, the number and sequence of plies and the temperature on the critical axial load and vibration frequency parameter within SDT and Kirchhoff–Love theory (KLT) are presented with numerical examples.

RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects

This article discusses standard Radial Basis Function (RBF) partition of unity method and an enhanced version known as the direct RBF partition of unity method. We introduce the framework of these methods for the effective and accurate numerical analysis of certain two-dimensional time-dependent Partial Differential Equations (PDEs). Additionally, we conduct a convergence analysis of these methods and establish error bounds for local approximations. These error bounds are determined based on conditions associated with the eigenvalues of the Laplacian operator matrices generated by the proposed methods. To validate the accuracy and reliability of these approaches, we provide several numerical examples. We assess the consistency between theoretical and numerical results. The localized techniques introduced in this research demonstrate a significant reduction in computational cost and algorithmic complexity, achieved through the direct local procedure. Despite this computational advantage, both versions of the RBF partition of unity method exhibit a similar level of accuracy. Finally, we conduct a comparative analysis between the RBF partition of unity local methods and the RBF-generated finite differences method, offering valuable insights into the strengths and limitations of each approach.

An augmented subspace based adaptive proper orthogonal decomposition method for time dependent partial differential equations

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. We use the difference between the approximation obtained in the augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to construct the augmented subspaces, the residual type augmented subspace and the coarse-grid approximation type augmented subspace. We apply our new methods to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that both the residual type augmented subspace based adaptive POD method and the coarse-grid approximation type augmented subspace based adaptive POD method are more efficient than the existing adaptive POD methods, especially for the advection dominated models.

RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects

This article discusses standard Radial Basis Function (RBF) partition of unity method and an enhanced version known as the direct RBF partition of unity method. We introduce the framework of these methods for the effective and accurate numerical analysis of certain two-dimensional time-dependent Partial Differential Equations (PDEs). Additionally, we conduct a convergence analysis of these methods and establish error bounds for local approximations. These error bounds are determined based on conditions associated with the eigenvalues of the Laplacian operator matrices generated by the proposed methods. To validate the accuracy and reliability of these approaches, we provide several numerical examples. We assess the consistency between theoretical and numerical results. The localized techniques introduced in this research demonstrate a significant reduction in computational cost and algorithmic complexity, achieved through the direct local procedure. Despite this computational advantage, both versions of the RBF partition of unity method exhibit a similar level of accuracy. Finally, we conduct a comparative analysis between the RBF partition of unity local methods and the RBF-generated finite differences method, offering valuable insights into the strengths and limitations of each approach.

Fisher–KPP-type models of biological invasion: open source computational tools, key concepts and analysis

This review provides open-access computational tools that support a range of mathematical approaches to analyse three related scalar reaction–diffusion models used to study biological invasion. Starting with the classic Fisher–Kolomogorov (Fisher–KPP) model, we illustrate how computational methods can be used to explore time-dependent partial differential equation (PDE) solutions in parallel with phase plane and regular perturbation techniques to explore invading travelling wave solutions moving with dimensionless speed c ≥ 2 . To overcome the lack of a well-defined sharp front in solutions of the Fisher–KPP model, we also review two alternative modelling approaches. The first is the Porous–Fisher model where the linear diffusion term is replaced with a degenerate nonlinear diffusion term. Using phase plane and regular perturbation methods, we explore the distinction between sharp- and smooth-fronted invading travelling waves that move with dimensionless speed c ≥ 1 / 2 . The second alternative approach is to reformulate the Fisher–KPP model as a moving boundary problem on 0 < x < L ( t ) , leading to the Fisher–Stefan model with sharp-fronted travelling wave solutions arising from a PDE model with a linear diffusion term. Time-dependent PDE solutions and phase plane methods show that travelling wave solutions of the Fisher–Stefan model can describe both biological invasion ( c > 0 ) and biological recession ( c < 0 ) . Open source Julia code to replicate all computational results in this review is available on GitHub; we encourage researchers to use this code directly or to adapt the code as required for more complicated models.

Leveraging physics-informed neural computing for transport simulations of nuclear fusion plasmas

For decades, plasma transport simulations in tokamaks have used the finite difference method (FDM), a relatively simple scheme to solve the transport equations, a coupled set of time-dependent partial differential equations. In this FDM approach, typically over O(105) time steps are needed for a single discharge, to mitigate numerical instabilities induced by stiff transport coefficients. It requires significant computing time as costly transport models are repeatedly called in a serial manner, proportional to the number of time steps. Additionally, the unidirectional calculations of FDM make it difficult to predict regions prior to the initial condition or apply additional temporal constraints. In this study, we discuss using a new scheme to solve plasma transport based on physics-informed neural networks (PINNs). PINN iteratively updates a function that maps spatiotemporal coordinates to plasma states, gradually reducing errors in transport equations. The required number of updates in PINNs is several orders of magnitude less than the chronological iterations in FDM. Furthermore, it is free from numerical instabilities arising from finite grids and enables more versatile semi-predictive simulations with arbitrary spatiotemporal constraints. In this paper, we discuss the features and potentials of the tokamak transport solver using PINNs through comparisons with FDM, and also its drawbacks and challenges.

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

An Efficient Replacement for Schrödinger's Partial Differential Equation

The classical finite difference method for solving time-dependent partial differential equations has become quite tedious and requires the use of off-the-shelf algorithms such as those in MATLAB. The treatment by finite difference method then solution of the n resulting first order algebraic equations is quite difficult since the underlying matrix is singular. We propose an alternative revolutionary statistical technique using Bmatrix chains, which are a product of the Cairo techniques. This technique is valid for both classical macroscopic physics such as the heat diffusion equation and modern microscopic quantum mechanics such as Schrödinger's PDE. He completely neglects partial differential equations as if they never existed. The numerical results of the proposed numerical statistical method show stability, accuracy and superiority over conventional PDE methods.

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.

GRINN: a physics-informed neural network for solving hydrodynamic systems in the presence of self-gravity

Modeling self-gravitating gas flows is essential to answering many fundamental questions in astrophysics. This spans many topics including planet-forming disks, star-forming clouds, galaxy formation, and the development of large-scale structures in the Universe. However, the nonlinear interaction between gravity and fluid dynamics offers a formidable challenge to solving the resulting time-dependent partial differential equations (PDEs) in three dimensions (3D). By leveraging the universal approximation capabilities of a neural network within a mesh-free framework, physics informed neural networks (PINNs) offer a new way of addressing this challenge. We introduce the gravity-informed neural network (GRINN), a PINN-based code, to simulate 3D self-gravitating hydrodynamic systems. Here, we specifically study gravitational instability and wave propagation in an isothermal gas. Our results match a linear analytic solution to within 1% in the linear regime and a conventional grid code solution to within 5% as the disturbance grows into the nonlinear regime. We find that the computation time of the GRINN does not scale with the number of dimensions. This is in contrast to the scaling of the grid-based code for the hydrodynamic and self-gravity calculations as the number of dimensions is increased. Our results show that the GRINN computation time is longer than the grid code in one- and two- dimensional calculations but is an order of magnitude lesser than the grid code in 3D with similar accuracy. Physics-informed neural networks like GRINN thus show promise for advancing our ability to model 3D astrophysical flows.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

Positional embeddings for solving PDEs with evolutional deep neural networks

This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on eigenfunctions of the Laplace-Beltrami operator, geometric properties are encoded intrinsically and Dirichlet, Neumann and periodic boundary conditions of the PDE solution are enforced directly through the neural network architecture. The proposed embeddings lead to improved error convergence for static PDEs and extend EDNNs towards computational domains of realistic complexity.Several steps are taken to improve performance of EDNNs: Solving the EDNN update equation with a Krylov solver avoids the explicit assembly of Jacobians and enables scaling to larger neural networks. Computational efficiency is further improved by an ad-hoc active sampling scheme that uses the PDE dynamics to effectively sample collocation points. A modified linearly implicit Rosenbrock method is proposed to alleviate the time step requirements of stiff PDEs. Lastly, a completely training-free approach, which automatically enforces initial conditions and only requires time integration, is compared against EDNNs that are trained on the initial conditions. We report results for the Korteweg-de Vries equation, a nonlinear heat equation and (nonlinear) advection-diffusion problems on domains with and without holes and various boundary conditions, to demonstrate the effectiveness of the method. The numerical results highlight EDNNs as a promising surrogate model for parametrized PDEs with slow decaying Kolmogorov n-width.

Modeling viscous dissipation in MHD boundary layer flow: A two-stage in time and compact in space finite difference approach

It is suggested that an explicit-implicit third-order in-time numerical scheme be used to solve time-dependent partial differential equations (PDEs). A fourth-order compact scheme in space is adopted for discretizing space variables. The work creatively blends a third-order explicit-implicit numerical scheme in time with a fourth-order compact scheme in space to solve time-dependent PDEs. Notably, the proposed numerical scheme demonstrates stability for scalar PDEs and exhibits convergence for both linear and nonlinear PDE systems. In addition, a mathematical model is given for the heat transfer of magnetohydrodynamic (MHD) flow over flat and oscillatory sheets, and the effect of viscous dissipation is also considered. For various governing parameters, features of the flow and heat transfer characteristics are discussed and examined. Some results are verified using an exact solution. We use an explicit-implicit finite difference method guaranteed to be stable and convergent under all conditions to solve the coupled nonlinear partial differential equations that describe this problem. Physical aspects of the problem are discussed in light of the system’s embedded parameters, and numerical and graphical results are presented. To sum up, progress in our knowledge of such systems relies heavily on creating an accurate and efficient numerical scheme for modeling MHD boundary layer flows while considering the influence of viscous dissipation. This study aims to improve the current state of MHD flow simulations and provide a useful resource for engineers and scientists engaged in developing and improving MHD-based technologies.

Neural Oscillators for Generalization of Physics-Informed Machine Learning

A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims to enhance the generalization capabilities of PIML, facilitating practical, real-world applications where accurate predictions in unexplored regions are crucial. We leverage the inherent causality and temporal sequential characteristics of PDE solutions to fuse PIML models with recurrent neural architectures based on systems of ordinary differential equations, referred to as neural oscillators. Through effectively capturing long-time dependencies and mitigating the exploding and vanishing gradient problem, neural oscillators foster improved generalization in PIML tasks. Extensive experimentation involving time-dependent nonlinear PDEs and biharmonic beam equations demonstrates the efficacy of the proposed approach. Incorporating neural oscillators outperforms existing state-of-the-art methods on benchmark problems across various metrics. Consequently, the proposed method improves the generalization capabilities of PIML, providing accurate solutions for extrapolation and prediction beyond the training data.

SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks

Dynamics of many complex systems, from weather and climate to spread of infectious diseases, can be described by partial differential equations (PDEs). Such PDEs involve unknown function(s), partial derivatives, and typically multiple independent variables. The traditional numerical methods for solving PDEs assume that the data are observed on a regular grid. However, in many applications, for example, weather and air pollution monitoring delivered by the arbitrary located weather stations of the National Weather Services, data records are irregularly spaced. Furthermore, in problems involving prediction analytics such as forecasting wildfire smoke plumes, the primary focus may be on a set of irregular locations associated with urban development. In recent years, deep learning (DL) methods and, in particular, graph neural networks (GNNs) have emerged as a new promising tool that can complement traditional PDE solvers in scenarios of the irregular spaced data, contributing to the newest research trend of physics informed machine learning (PIML). However, most existing PIML methods tend to be limited in their ability to describe higher dimensional structural properties exhibited by real world phenomena, especially, ones that live on manifolds. To address this fundamental challenge, we bring the elements of the Hodge theory and, in particular, simplicial convolution defined on the Hodge Laplacian to the emerging nexus of DL and PDEs. In contrast to conventional Laplacian and the associated convolution operation, the simplicial convolution allows us to rigorously describe diffusion across higher order structures and to better approximate the complex underlying topology and geometry of the data. The new approach, Simplicial Neural Networks for Partial Differential Equations (SNN PDE) offers a computationally efficient yet effective solution for time dependent PDEs. Our studies of a broad range of synthetic data and wildfire processes demonstrate that SNN PDE improves upon state of the art baselines in handling unstructured grids and irregular time intervals of complex physical systems and offers competitive forecasting capabilities for weather and air quality forecasting.

Neural Physical Simulation with Multi-Resolution Hash Grid Encoding

We explore the generalization of the implicit representation in the physical simulation task. Traditional time-dependent partial differential equations (PDEs) solvers for physical simulation often adopt the grid or mesh for spatial discretization, which is memory-consuming for high resolution and lack of adaptivity. Many implicit representations like local extreme machine or Siren are proposed but they are still too compact to suffer from limited accuracy in handling local details and a long time of convergence. We contribute a neural simulation framework based on multi-resolution hash grid representation to introduce hierarchical consideration of global and local information, simultaneously. Furthermore, we propose two key strategies: 1) a numerical gradient method for computing high-order derivatives with boundary conditions; 2) a range analysis sample method for fast neural geometry boundary sampling with dynamic topologies. Our method shows much higher accuracy and strong flexibility for various simulation problems: e.g., large elastic deformations, complex fluid dynamics, and multi-scale phenomena which remain challenging for existing neural physical solvers.

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.

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

Proper orthogonal decomposition method of constructing a reduced-order model for solving partial differential equations with parametrized initial values

This paper proposes a novel proper orthogonal decomposition method for constructing a reduced-order model. This model effectively computes solutions for various initial conditions in time-dependent partial differential equations. The mode obtained from the proposed proper orthogonal decomposition captures both the fluctuating component and the mean-field of the time-dependent solution. In this mode, the eigenvalues representing the mean-field, are significantly larger than those representing the fluctuating component. Consequently, determining the number of modes required for representing a solution cannot rely solely on the cumulative contribution rate. Therefore, the proposed method gives a scalar weight to the mean-field and controls the magnitude of the mean-field energy. We propose a method that considers the magnitude of the scaler weight alongside the cumulative contribution ratio for determining the number of modes. This method was evaluated using the time-dependent Burgers equation with parametric initial values. Our proposed method selects the appropriate mode to represent the given dataset, unlike the conventional method. A reduced-order model based on the proposed method effectively computes the time evolutions of the solutions using datasets for several parameters by imposing a condition that the cumulative contribution ratio exceeds 98%.

A fast computational technique to solve fourth-order parabolic equations: application to good Boussinesq, Euler-Bernoulli and Benjamin-Ono equations

In this article, we introduce a novel cubic spline method for the numerical solution of a class of fourth-order time-dependent parabolic partial differential equations. The method employs uniform mesh discretization, achieving a spatial accuracy of four and a temporal accuracy of two. Here, the key contribution is that this method can be directly applied to singular parabolic problems due to the consideration of half-step points in spatial direction. The model linear fourth-order partial differential equation is used to discuss the method's stability. The method we propose has multiple advantages, including high accuracy and seamless point-to-point interpolation capabilities. It is a versatile technique that can easily adapt to solve various nonlinear problems without the need to change or linearize the nonlinear terms to produce numerical solutions. Benchmark linear and nonlinear problems like single and double-soliton of the good Boussinesq equation, the Euler-Bernoulli beam model and the Benjamin-Ono equation, are solved.