Methods For Partial Differential Equations Research Articles

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors

Abstract A gradient discretisation method (GDM) is an abstract setting that designs the unified convergence analysis of several numerical methods for partial differential equations and their corresponding models. In this paper, we study the GDM for anisotropic reaction–diffusion problems, based on a general reaction term, with Neumann boundary condition. With natural regularity assumptions on the exact solution, the framework enables us to provide proof of the existence of weak solutions for the problem, and to obtain a uniform-in-time convergence for the discrete solution and a strong convergence for its discrete gradient. It also allows us to apply non-conforming numerical schemes to the model on a generic grid (the non-conforming ℙ ⁢ 1 {\mathbb{P}1} finite element scheme and the hybrid mixed mimetic (HMM) methods). Numerical experiments using the HMM method are performed to assess the accuracy of the proposed scheme and to study the growth of glioma tumors in heterogeneous brain environment. The dynamics of their highly diffusive nature is also measured using the fraction anisotropic measure. The validity of the HMM is examined further using four different mesh types. The results indicate that the dynamics of the brain tumor is still captured by the HMM scheme, even in the event of a highly heterogeneous anisotropic case performed on the mesh with extreme distortions.

Helmholtz decomposition-based SPH

BackgroundSPH method has been widely used in the simulation of water scenes. As a numerical method of partial differential equations, SPH can easily deal with the distorted and complex boundary. In addition, the implementation of SPH is relatively simple, and the results are stable and not easy to diverge. However, SPH method also has its own limitations. In order to further improve the performance of SPH method and expand its application scope, a series of key and difficult problems restricting the development of SPH need to be improved. MethodsIn this paper, we introduce the idea of Helmholtz decomposition into the framework of smoothed particle hydrodynamics (SPH) and propose a novel velocity projection scheme for three-dimensional water simulation. First, we apply Helmholtz decomposition to a three-dimensional velocity field and decompose it into three orthogonal subspaces. Then, our method combines the idea of spatial derivatives in SPH to obtain a discrete Poisson velocity equation. Finally, the conjugate gradient (CG) is utilized to efficiently solve the Poisson equation. ResultsThe experimental results show that the proposed scheme is suitable for various situations and has higher efficiency than the current SPH projection scheme. ConclusionCompared with the previous projection scheme, our solution does not need to modify the particle velocity indirectly by pressure projection, but directly by velocity field projection. The new scheme can be well integrated into the existing SPH framework, and can be applied to the interaction of water with static and dynamic obstacles, even for viscous fluid.

Two‐sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems

AbstractNumerical methods for elliptic partial differential equations usually lead to systems of linear equations with sparse, symmetric, and positive definite matrices. In many methods, these matrices can be obtained as sums of local symmetric positive semidefinite matrices. In this article, we use this assumption and introduce a method that provides guaranteed lower and upper bounds to all individual eigenvalues of the preconditioned matrices. We apply the method for preconditioners arising from the same discretization problem but with simplified coefficients. The method uses solely the data over the solution domain and local connections between the degrees of freedom defined by the discretization.

Chebyshev wavelet collocation method for magnetohydrodynamic flow equations

This study proposes Chebyshev wavelet collocation method for partial differential equation and applies to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Approximate solutions of velocity and induced magnetic field are obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. Numerical results of the MHD flow problem show that the accuracy of proposed method is quite good even in the case of a small number of grid points. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number Ha ≤ 1000.

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

Nonlinear methods for model reduction

Typical model reduction methods for parametric partial differential equations construct a linear space Vn which approximates well the solution manifold M consisting of all solutions u(y) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n-term approximation, and certain tree-based methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space Vn by a nonlinear space Σn. Little is known in terms of their performance guarantees, and most existing numerical experiments use a parameter dimension of at most two. In this work, we make a step towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation, we give a first comparison of their performance with the performance of standard linear approximation for any compact set. We then study these methods for solution manifolds of parametrized elliptic PDEs. We study a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and give performance guarantees with respect to accuracy of approximation versus m and N.

Exact solutions of non-linear Klein–Gordon equation with non-constant coefficients through the trial equation method

In this note, we use an extension of the trial equation method (also called the direct integral method) for partial differential equations with non-constant coefficients to derive exact solutions in the form of nonlinear waves. The model considered generalizes other classical models from physics like the Klein–Gordon equation, the $$(1 + 1)$$ -dimensional $$\phi ^4$$ -theory, the Fisher–Kolmogorov equation from population dynamics, and the Hodgkin–Huxley model which describes the propagation of electrical signals through the nervous system. As a particular example, the cylindrically symmetric cubic nonlinear Klein–Gordon equation is considered herein.

Nested Monte Carlo simulation in financial reporting: a review and a new hybrid approach

Risk assessment on a stochastic basis has become prevalent in financial reporting due to increasingly sophisticated regulatory requirements. Many applications require nested stochastic projections, for which crude Monte Carlo can be too costly and time-consuming to perform to reach a reasonable degree of accuracy. While there has been ample literature on nested simulation methods in the area of portfolio management, less is known in the literature regarding nested simulation for financial reporting and internal risk management. There has been little research dealing with unique challenges arising from the structure of insurance liabilities. This paper intends to fill the gap in the literature by providing an overview of the use of nested stochastic modeling for different regulatory purposes and investigating the multi-period nested stochastic model, common in insurance products. The paper reviews a variety of so-called ‘stochastic-on-stochastic’ methods to speed up nested simulations. In addition, the paper presents a new hybrid ‘deterministic-on-stochastic’ method based on partial differential equation (PDE). To the best knowledge of the authors, this is the first time that a PDE method has been introduced for the purpose of nested stochastic projection. A numerical example is provided to show the high efficiency of the hybrid PDE method in a multi-period nested model for financial reporting.

Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of bounded Riesz systems and generalizes random partial Fourier matrices. Our main result improves the currently available results for the null space and restricted isometry properties of such random matrices. The main novelty of our analysis is a new upper bound for the expectation of the supremum of a Bernoulli process associated with a restricted isometry constant. We apply our result to prove new performance guarantees for the CORSING method, a recently introduced numerical approximation technique for partial differential equations (PDEs) based on compressive sensing.

A stabilizer free weak Galerkin finite element method for parabolic equation

In this article, a stabilizer free weak Galerkin (SFWG) finite element method for a parabolic partial differential equation is proposed. The goal of using the SFWG method is to make the numerical implementation easier and more efficient. The optimal rate of convergence is derived in both H1 and L2 norms. Numerical experiments are performed to verify the theoretical results.

Time delay induced Hopf bifurcation in a diffusive predator\u2013prey model with prey toxicity

In this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

A Study on Finite Difference Explicit Numerical Method of Symmetric Hyperbolic Partial Differential Equation

Explicit method approaches numerical analysis for obtaining numerical approximation to the solution in time dependent partial differential equation and ordinary differential equation which is required in computer application and physical processes. Mainly explicit methods are used to calculate the system at current time. Explicit method are very simple but not stable. Mainly in explicit numerical methods, parameters are calculated based on previous levels and also explicit solutions are independent to other values. For example the unsteady heat equation can be solved explicitly. The present paper is concerned with explicit numerical method of symmetric hyperbolic partial differential equation by using constant indices and also we proved the monotonical preserving values and also discussed the symmetrical hyperbolic partial differential equation with constant coefficients.

Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Spatial Gradient of the Height

The aim of this investigation is to determine the effect of fluid leak-off (suction) and fluid injection (blowing) at the horizontal base on the two-dimensional spreading under the gravity of a thin film of viscous incompressible fluid by studying the evolution of the streamlines in the thin film. It is assumed that the normal component of the fluid velocity at the base is proportional to the spatial gradient of the height of the film. Lie symmetry methods for partial differential equations are applied. The invariant solution for the surface profile is derived. It is found that the thin fluid film approximation is satisfied for weak to moderate leak-off and for the whole range of fluid injection. The streamlines are derived and plotted by solving a cubic equation numerically. For fluid injection, there is a dividing streamline originating at the stagnation point at the base which separates the flow into two regions, a lower region consisting mainly of rising fluid and an upper region consisting mainly of descending fluid. An approximate analytical solution for the dividing streamline is derived. It generates an approximate V-shaped surface along the length of the two-dimensional film with the vertex of each section the stagnation point. It is concluded that the fluid flow inside the thin film can be visualised by plotting the streamlines. Other models relating the fluid velocity at the base to the height of the thin film can be expected to contain a dividing streamline originating at a stagnation point and dividing the flow into a lower region of rising fluid and an upper region of descending fluid.

Sparse spectral methods for partial differential equations on spherical caps

Abstract In recent years, sparse spectral methods for solving partial differential equations have been derived using hierarchies of classical orthogonal polynomials (OPs) on intervals, disks, disk-slices and triangles. In this work, we extend the methodology to a hierarchy of non-classical multivariate OPs on spherical caps. The entries of discretizations of partial differential operators can be effectively computed using formulae in terms of (non-classical) univariate OPs. We demonstrate the results on partial differential equations involving the spherical Laplacian and biharmonic operators, showing spectral convergence with discretizations that can be made well conditioned using a simple preconditioner.

Book Reviews

The section opens with a review well written by a giant in the field of numerical solution of partial differential equations, Roland Glowinski. In this featured review, he analyzes the book Think Before You Compute: A Prelude to Computational Fluid Dynamics, by E. J. Hinch, in much detail and contributes his own thoughts on appraisal or possibilities for improvement to every chapter of the book. The reviewer's verdict on the book is: “The book is a gem.” The following three reviewed books, written in an earlier and different world, take on a whole new relevance against the background of the current pandemic situation. First, we have the book by Irina Kareva and Georgy Karev, Modeling Evolution of Heterogeneous Populations: Theory and Applications. This book is diligently reviewed by Karen Page, who rates the book as “enlightening" even if it still contains small annoying errors. The following book, Delay-Adaptive Linear Control, by Yang Zhu and Miroslav Krstic, also seems to anticipate some aspects of the current modeling and control of the COVID-19 pandemic. This book is reviewed in detail by Bozenna Pasik-Duncan and is classified as an important contribution to control in delay systems. For the classification and mining of numerous medical data, such as those being generated right now, Mehrshad Sadria and Anita Layton suggest reading Bioinformatics, edited by Andreas D. Baxevanis, Gary D. Bader, and David S. Wishart, and recommend it as an “easy to read, greatly informative, and highly timely” book. Finally, I would like to mention the book by Zhendong Luo and Goong Chen on Proper Orthogonal Decomposition Methods for Partial Differential Equations, which is reviewed by the expert Stefan Volkwein, who praises the book as providing “a really good recipe for the application of the POD method to many nonstationary PDEs,” and also the book Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno, by Alexander J. Hahn, that allows the reader to escape the current narrow earthly confinement in mathematical thoughts and delve into interplanetary explorations.

Multilevel Quasi Monte Carlo Methods for Elliptic PDEs with Random Field Coefficients via Fast White Noise Sampling

When solving partial differential equations (PDEs) with random fields as coefficients, the efficient sampling of random field realizations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the stochastic PDE (SPDE) approach of Lindgren et al. combined with a new fast algorithm for white noise sampling which is tailored to (ML)QMC. We express white noise as a wavelet series expansion that we divide into two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good quasi Monte Carlo (QMC) convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction, yielding an overall linear cost. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.

Spectral collocation method for stochastic partial differential equations with fractional Brownian motion

In this paper, we consider the numerical approximation of stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with Hurst index H>12. A Fourier spectral collocation approximation is used in space and semi-implicit Euler method is applied for the temporal approximation. Our aim is to investigate the convergence of the proposed method. Optimal strong convergence error estimates in mean-square sense are derived and numerical experiments are presented and confirm theoretical results.