Methods For Partial Differential Equations Research Articles

The virtual element method

The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very general shape.Nonetheless, the paper is also addressed to readers who have already heard (and possibly read) about VEMs and are interested in gaining more precise information, in particular concerning their application in specific subfields such as ${C}^1$ -approximations of plate bending problems or approximations to problems in solid and fluid mechanics.

A semi-analytical dynamics method for spindle radial throw in boring process

In precision boring processing, accurate prediction of the radial throw of a boring tool spindle has critical significance for reducing machining errors. However, the classical dynamics method of radial throw based on the rigid body partition method ignores special characteristics of the boring tool spindle, resulting in additional errors regarding radial throw evaluation. Hence, a semi-analytical dynamics method considering the axial length, variable cross-section, and offset distance was proposed to decrease the computational error of the current dynamics model. The mechanism of radial throw was analyzed by the mechanics of materials and mathematical formulas. The Euler-Bernoulli beam theory and partial differential equation method were utilized to derive the dynamics model of the boring tool spindle, and an improved Galerkin variational algorithm was used to resolve the dynamics formulas for the radial throw. Then, the proposed dynamics method was compared with the non-contact displacement measurement and the traditional lumped mass parameter method. The validated results showed that the presented dynamics method had good agreement with the experimental data, and the maximum predictive error was 16.08 μm lower based on the proposed method. Furthermore, the machining precision of holes was also clearly evaluated and discussed using the proposed dynamics method under different processing conditions, and the lower uncertainty of the proposed dynamics method was also demonstrated. As a result, the proposed semi-analytical dynamics method can accurately compute the radial throw and evaluate the machining error in boring technology.

Canonical Euler splitting method for parabolic partial functional differential algebraic equations

A novel canonical Euler splitting method is presented for semilinear composite stiff parabolic partial functional differential algebraic equations with initial and Dirichlet boundary conditions. The original partial differential problems are transformed into the semi-discrete problems by spatial discretization, and then the canonical Euler splitting method is employed to solve the resulting semi-discrete problems. Under appropriate assumptions, the stability and convergence theories of this method are established. A series of numerical experiments are given to illustrate the effectiveness of this method and the correctness of theoretical results.Numerical results also demonstrate that the constructed method can significantly improve the calculation speed.

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments

AbstractIn this paper, the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments are investigated. Firstly, the variation formulation is derived by applying Green's formula and Galerkin finite element method to spatial direction of the original equation. Next, semidiscrete and fully discrete schemes are obtained and the convergence is analyzed in ‐norm rigorously. Moreover, the stability analysis shows that the semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are also obtained under which the analytic solution is asymptotically stable. Finally, some numerical experiments are provided to demonstrate our theoretical results.

Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

In this paper, we propose new geometrically unfitted space-time finite element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretization in time, we consider discontinuous Galerkin, as well as related continuous (Petrov–)Galerkin and Galerkin collocation methods. For stabilization with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilization is employed. The article puts an emphasis on the techniques that allow us to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.

A Mingled Tau-Finite Difference Method for Stochastic First-Order Partial Differential Equations

A Mingled Tau-Finite Difference Method for Stochastic First-Order Partial Differential Equations

A posteriori error estimates of two-grid weak Galerkin methods for semilinear elliptic differential equations

In this paper, we investigate the residual-based a posteriori error estimates of two-grid weak Galerkin (WG) methods for second order semilinear elliptic partial differential equations (PDEs). First, we propose two different two-grid weak Galerkin methods for the model problem and then establish a posteriori error estimators of the two-grid weak Galerkin methods. Theoretical analysis is given to prove the reliability and efficiency of the error estimators. We mainly study the lowest order case of WG element (Pj(T),Pℓ(∂T),RTj(T)) with j=ℓ=0[19]. Numerical experiments are provided to confirm the theoretical results.

Fourier-Transform Method for Partial Differential Equations. Part 2. Existence and Uniqueness of Solutions to the Cauchy Problem for Linear Equations

Fourier-Transform Method for Partial Differential Equations. Part 2. Existence and Uniqueness of Solutions to the Cauchy Problem for Linear Equations

Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Inhomogeneous Boundary Value Problems with High Contrast Coefficients

In this article we develop the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for elliptic partial differential equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions, and the high contrast property emerges from the coefficients of elliptic operators and Robin boundary conditions. By careful construction of multiscale bases of the CEM-GMsFEM, we introduce two operators and which are used to handle inhomogeneous Dirichlet and Neumann boundary values and are also proved to converge independently of contrast ratios as enlarging oversampling regions. We providean a priori error estimate and show that the number of oversampling layers is the key factor in controlling numerical errors. A series of experiments are conducted, and those results reflect the reliability of our methods even with high contrast ratios.

An h-Adaptive Poly-Sinc-Based Local Discontinuous Galerkin Method for Elliptic Partial Differential Equations

For the purpose of solving elliptic partial differential equations, we suggest a new approach using an h-adaptive local discontinuous Galerkin approximation based on Sinc points. The adaptive approach, which uses Poly-Sinc interpolation to achieve a predetermined level of approximation accuracy, is a local discontinuous Galerkin method. We developed an a priori error estimate and demonstrated the exponential convergence of the Poly-Sinc-based discontinuous Galerkin technique, as well as the adaptive piecewise Poly-Sinc method, for function approximation and ordinary differential equations. In this paper, we demonstrate the exponential convergence in the number of iterations of the a priori error estimate derived for the local discontinuous Galerkin technique under the condition that a reliable estimate of the precise solution of the partial differential equation at the Sinc points exists. For the purpose of refining the computational domain, we employ a statistical strategy. The numerical results for elliptic PDEs with Dirichlet and mixed Neumann-Dirichlet boundary conditions are demonstrated to validate the adaptive greedy Poly-Sinc approach.

An adaptive wavelet method for nonlinear partial differential equations with applications to dynamic damage modeling

Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet-based technique to solve a coupled system of nonlinear partial differential equations (PDEs) while resolving features on a wide range of spatial and temporal scales. The algorithm exploits the multiresolution nature of wavelet basis functions to solve initial-boundary value problems on finite domains with a sparse multiresolution spatial discretization. By leveraging wavelet theory and embedding a predictor-corrector procedure within the time advancement loop, we dynamically adapt the computational grid and maintain accuracy of the solutions of the PDEs as they evolve. Consequently, our method provides high fidelity simulations with significant data compression. We present verification of the algorithm and demonstrate its capabilities by modeling high-strain rate damage nucleation and propagation in nonlinear solids using a novel Eulerian-Lagrangian continuum framework.

Machine-learning-based spectral methods for partial differential equations

Spectral methods are an important part of scientific computing’s arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet) to identify candidate functions on which to expand the solution of PDEs. We have devised an approach that uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions that have the following properties: (1) they constitute a basis, (2) they are orthonormal, and (3) they are hierarchical, i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs. The proposed approach advances the state of the art and versatility of spectral methods and, more generally, promotes the synergy between traditional scientific computing and machine learning.

One-stage explicit trigonometric integrators for effectively solving quasilinear wave equations

In this paper, one-stage explicit trigonometric integrators for solving quasilinear wave equations are formulated and studied. For solving wave equations, we first introduce trigonometric integrators as the semidiscretization in time and then consider a spectral Galerkin method for the discretization in space. We show that one-stage explicit trigonometric integrators in time have second-order convergence and the result is also true for the fully discrete scheme without requiring any CFL-type coupling of the discretization parameters. The results are proved by using energy techniques, which are widely applied in the numerical analysis of methods for partial differential equations.

Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution to a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.

Space-time error estimates for deep neural network approximations for differential equations

Over the last few years deep artificial neural networks (ANNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form [0,T]times mathbb {R}^{d}ni (t,x){kern -.5pt}mapsto {kern -.5pt} tx{kern -.5pt}in {kern -.5pt} mathbb {R}^{d} where T{kern -.5pt}in {kern -.5pt} (0,infty ), d{kern -.5pt}in {kern -.5pt} mathbb {N}, which we both develop within this article.

Stability and convergence of a new hybrid method for fractional partial differential equations

Stability and convergence of a new hybrid method for fractional partial differential equations

Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations

The article proposes a method for analyzing the Cauchy problem for a wide class of evolutionary linear partial differential equations with variable coefficients. By applying the (inverse) Fourier transform, the original equation is reduced to an integro-differential equation, which can be considered as an ordinary differential equation in the corresponding Banach space. The selection of this space is carried out in such a way that the principle of contraction mappings can be used. To carry out the corresponding estimates for the operators generated by the transformed equation, we impose the conditions of finiteness in the space variable for the inverse Fourier transform of the coefficients, and the spaces of the coefficients of the original equation are determined from the Paley-Wiener Fourier transform theorems. In this case, the apparatus of the theory of the Bochner integral in pseudo-normed spaces, countably-normed spaces and Sobolev spaces is used. Classes of functions are distinguished in which the existence and uniqueness of solutions are proved. For equations with coefficients with separated variables, exact solutions are obtained in the form of a Fourier transform of finite sums for operator exponentials.

An overview on deep learning-based approximation methods for partial differential equations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.

Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility

<abstract><p>In general, derivation of closed-form analytic formulas for the prices of path-dependent exotic options is a challenging task when the underlying asset price model is chosen to be a stochastic volatility model. Pricing stop-loss and Russian options is studied under a multiscale stochastic volatility model in this paper. Both options are commonly perpetual American-style derivatives with a lookback provision. We derive closed-form formulas explicitly for the approximate prices of these two exotic options by using multiscale asymptotic analysis and partial differential equation method. The formulas can be efficiently computed starting with the Black-Scholes option prices. The accuracy of the analytic approximation is verified via Monte-Carlo simulations and the impacts of the multiscale stochastic volatility on the corresponding Black-Scholes option prices are revealed. Also, the performance of the model is compared with that of other models.</p></abstract>

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations

This paper proposes a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic partial differential equations (PDEs). The underlying new computational method implements the method of so-called approximating fuzzy components, which evaluate the solution values with fourth-order accuracy at internal mesh points. Triangular basic functions and fuzzy components are locally determined by linear combinations of solution values at nine points. Such a scheme connects the proposed method of approximating fuzzy components with the exact values of the solution using a linear system of equations. Compact approximations of high-resolution fuzzy components using nine points give a block tridiagonal Jacobi matrix. Apart from the numerical solution, it is easy to construct closed-form approximate solutions using a 2D spline interpolation polynomial from the available data with fuzzy components. The upper bounds of the approximation errors are estimated, as well as the convergence of the approximating solutions. Simulations with linear and nonlinear elliptical PDEs arising from quantum mechanics and convection-dominated diffusion phenomena are presented to confirm the usefulness of the new scheme and fourth-order convergence. To summarize:•The paper presents a high-resolution numerical method for the two-dimensions elliptic PDEs with nonlinear terms.•The combined effect of fuzzy transform and compact discretizations yields almost fourth-order accuracies to Schro¨dinger equation, convection-diffusion equation, and Burgers equation.•The high-order numerical scheme is computationally efficient and employs minimal data storage.