Petrov-Galerkin Method Research Articles

Computational approaches on integrating vaccination and treatment strategies in the SIR model using Galerkin time discretization scheme

ABSTRACT In this manuscript, we carried out a thorough analysis of the general SIR model for epidemics. We broadened the model to include vaccination, treatment, and incidence rate. The vaccination rate is a testament to the alternatives made by individuals when it comes to receiving vaccinations and merging with the community of the recovered. The treatment rate measures how often people who have contracted a disease are able to transition into the recovered category. The continuous Galerkin-Petrov method, specifically the cGP(2)-method, is employed to calculate the numerical solutions of the models. The cGP(2)-method imperative to analyse two unknowns over every time interval. The unknowns can be determined by solving a block system of size (2 × 2). This approach demonstrates a strong level of precision throughout the entire time interval, with an impressive rate of convergence at the discrete time points. In addition, the article investigates the concept of the basic reproduction number ( R 0 ) and explores the intricacies of conducting sensitivity analysis of the developed system.

A Modified Bubnov-Galerkin Method for Solving Boundary Value Problems with Linear Ordinary Differential Equations

Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation. Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions. Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order. Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.

A stable localized weak strong form radial basis function method for modelling variably saturated groundwater flow induced by pumping and injection

The unsaturated zone profoundly affects groundwater (GW) flow induced by pumping and injection due to the capillary forces. However, current radial basis function (RBF) numerical models for GW pumping and injection mostly ignore the unsaturated zone. To bridge this gap, we developed a new three-dimensional weak strong form RBF model in this study, called CCHE3D-GW-RBF. Flow processes were modelled by the mixed-form Richards equation which was iteratively solved by the modified Picard iteration. Soil-water characteristic curves were represented by the widely applicable formulas, the van Genuchten (1980) model. Differential operators were approximated by the localized Gaussian RBF, and the weighted singular value decomposition method was used to construct stable bases. The injection/pumping wells and the flux boundaries were handled by the weak formulation using Meshless Local Petrov Galerkin method, and the strong-form equation using the collocation RBF method was enforced on the other points. Good agreement was found between the simulation results from our numerical model and the well-accepted solutions in all three verification cases, demonstrating the accuracy and applicability of this model. In addition, a smaller RBF shape parameter was found to produce a more accurate modelling resulting, indicating the necessity of implementing stable bases for RBF models.

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media

An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media

Fractional non-Fourier modeling of laser drilling process

In this paper, a novel fractional non-Fourier model is employed to simulate the laser drilling process, addressing limitations inherent in classical heat conduction equations, including the well-known heat equation paradox associated with infinite heat propagation velocity. This model approach combines spatial approximation via the Meshless Local Petrov-Galerkin method with temporal approximation using the Grünwald-Letnikov finite difference scheme. The study assesses the impact of employing fractional orders, both constant and variable over time, on numerical results, and validates the model using experimental data.

Time-dependent dynamical energy analysis via convolution quadrature

Dynamical Energy Analysis was introduced in 2009 as a novel method for predicting high-frequency acoustic and vibrational energy distributions in complex engineering structures. In this paper we introduce the first time-dependent Dynamical Energy Analysis method. Time-domain models are important in numerous applications including sound simulation in room acoustics, predicting shock-responses in structural mechanics and modelling electromagnetic scattering from conductors. The first step is to reformulate Dynamical Energy Analysis in the time-domain by means of a convolution integral operator. We are then able to employ the Convolution Quadrature method to provide a link between the previous frequency-domain implementations of Dynamical Energy Analysis and fully time-dependent solutions by means of the Z-transform. By combining a modified multistep Convolution Quadrature approach for the time discretisation, together with Galerkin and Petrov-Galerkin methods for the space and momentum discretisations, respectively, we are able to accurately track the propagation of high-frequency transient signals through phase-space. The implementation here is detailed for finite two-dimensional spatial domains and we demonstrate the versatility of our approach by performing a range of numerical experiments for regular, non-convex and irregular geometries as well as different types of wave source.

Spectral Petrov-Galerkin Method for Solving the Two-Dimensional Integral Equations of the Second Kind

Spectral Petrov-Galerkin Method for Solving the Two-Dimensional Integral Equations of the Second Kind

Hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations

<i>hp</i>-version <i>C</i>1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations

Corium interface flow dynamics investigation during severe accident in pressurized water reactors using compressive advection interface capturing method

Past nuclear accident occurrences raised strong concerns which led to research on nuclear safety. One of the major causes of nuclear accidents is the impeded circulation of core coolant, leading to decay heat removal cessation and rapid temperature rise. If uncontrolled, this results in critical heat flux, loss of coolant accidents, and core dryout. Detailed melted core relocation (i.e., nuclear fuel, graphite, and zircaloy) needs to be investigated through interface capture and multimaterial flow model coupling, which have not been done in previous studies. This work aims to investigate the impacts of temperature and core material composition on the flow dynamics during core relocation. In this study, mass fraction is discretized using a streamlined upwind Petrov–Galerkin method spatially and a modified Crank–Nicolson method temporally to accurately capture fluid interfaces using a high-order accurate flux-limiter. Two core material composition cases (individual material properties case and bulk material properties case) were considered to assess the impact of temperature and core materials composition on both flow dynamics and computational time. Temperature has a significant impact on core material transport and corium flow dynamics during core relocation. Bulk materials properties case has greater impact of temperature on its corium resulting in faster materials transport, but with higher computation time.

A Uniform Convergent Petrov-Galerkin Method for a Class of Turning Point Problems

A Uniform Convergent Petrov-Galerkin Method for a Class of Turning Point Problems

Disclosure of the intricacies in coupled groundwater flow and contaminant transport using mesh-less local Petrov–Galerkin method

Disclosure of the intricacies in coupled groundwater flow and contaminant transport using mesh-less local Petrov–Galerkin method

Multi-material topology optimization with a direct meshless local Petrov–Galerkin method applied to a two-dimensional boundary value problem

ABSTRACT This work presents a novel approach for the solution of multi-material topology optimization problems of elastic structures coupling the Direct Meshless Local Petrov–Galerkin (DMLPG) method with the Gradual Bi-direction Evolutionary Structural Optimization (gBESO) method. In recent years, meshless methods have been used in several engineering fields, showing some advantages compared to mesh-based methods. DMLPG is a truly meshless method, which achieves results with good accuracy and computational efficiency, avoiding the numerical integration of complicated shape functions by adopting low-degree polynomials. In the proposed algorithm, DMLPG is used to obtain smooth nodal displacements, strains and stresses, and gBESO updates the structural geometry based on design sensitivity values of the compliance while it gradually increases the Young's modulus of the material. The optimization problem minimizes the compliance objective function subject to volume constraints. Numerical examples demonstrate the applicability and validity of the technique.

A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations

Methods for stabilising the turbulent cascade of potential enstrophy are analysed and compared for a compatible finite element discretisation of the rotating shallow water equations. These different approaches to upwinding the potential vorticity include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a more recent method where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent. The SUPG scheme introduces a consistent correction to the APVM scheme that acts as a backscatter term ensuring a richer depiction of turbulent dynamics. The downwinded trial function formulation results in the advection of downwind corrections, which analysis and numerical experiments show to be quantitatively similar to the SUPG scheme for a turbulent shear flow. The main difference between the SUPG and downwinded trial function schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme.We also present new temporal formulations by which potential enstrophy is exactly integrated across each time level. Results using these formulations are observed to be stable in the absence of any dissipation, despite the aliasing of grid scale turbulence. Using such a formulation and the APVM with a coefficient O(100) times smaller that its regular value leads to turbulent spectra that are greatly improved at the grid scale over the SUPG and downwinded trial function formulations with unstable potential enstrophy errors.

Randomized Neural Networks with Petrov–Galerkin Methods for Solving Linear Elasticity and Navier–Stokes Equations

Randomized Neural Networks with Petrov–Galerkin Methods for Solving Linear Elasticity and Navier–Stokes Equations

Employing CNPS and CPS approaches to calculate numerical roots of ninth-order linear and nonlinear boundary value problems

The extremely accurate findings of ninth-order linear and nonlinear BVPs are described in this paper. CNPS and CPS are used to find out the numerical roots of linear and nonlinear, ninth-order BVPs. The proposed methods transform the ninth-order BVPs into a system of linear equations. The algorithms developed in this study not only provide approximate solutions to the BVPs using CNPS and CPS, but they also estimate the derivatives from the first to the ninth-order of the analytical solution simultaneously. These approaches are implemented across five diverse problems, showcasing the effectiveness of these techniques by utilizing step sizes of [Formula: see text] and [Formula: see text]. In order to gauge the accuracy of the methods, a comparison is drawn between the outcomes and the exact solutions, which are then presented in tabular and graphical format. The precision of these techniques is demonstrated through a detailed investigation and is found to be superior to the CBS, the Petrov–Galerkin Method (PGM) using Splines functions as basis functions and Septic B Splines functions as weight functions, the collocation method for ninth-order BVPs by Quintic B Splines and Sextic B Splines as evidenced by the comparison of AEs of CPS and CNPS with these alternative methods.

Error estimates of a space–time Legendre spectral method for solving the Korteweg–de Vries equation

In this paper, a space–time spectral method for solving the Korteweg–de Vries equation is considered. The discrete schemes of the method are based on the Legendre–Petrov–Galerkin method in spatial direction and the Legendre-tau method in temporal direction with nonperiodic boundary conditions. Stability analysis results and error estimates are obtained in L2-norm by introducing a cut-off function without Lipschitz condition. The method is also applicable to solve some (2m+1)th-order differential equations. Comparison of our numerical results with those of other spectral methods exhibits the accuracy of our methods for the Korteweg–de Vries equation.

Analysis and application of MLPG7 for diffusion equations with nonlinear reaction terms

This paper extends the recently proposed variant of meshless local Petrov Galerkin (MLPG) method, i.e., MLPG7, for solving time dependent PDEs. As test function, the method uses a novel modification of fundamental solution of Laplace operator that not only the test function itself but also its derivative vanish on boundary of local subdomains. Therefore, more stable local integral equations are obtained by replacing a boundary integral of derivatives of field variables with a domain integral of field variables itself. MLPG7 formulation converts the nonlinear governing equation into a system of first order ordinary differential equations (ODEs). For this system of ODEs, a theoretical stability analysis is carried out for the case of regularly spaced nodal points. Stability and convergence conditions for fully discretized equations by Euler (forward and backward) and Crank–Nicolson methods are investigated. The nonlinear term is treated iteratively. A numerical study investigates the convergence and stability of the method. A comparison is also done with some well-known methods and the results reveal the excellence of MLPG7. The convergence of the iterative approach is numerically verified by another illustrative test example. As nonlinear test problems, Allen–Cahn equations are studied.

A class of arbitrarily high-order energy-preserving method for nonlinear Klein–Gordon–Schrödinger equations

In this paper, we develop a class of arbitrarily high-order energy-preserving time integrators for the nonlinear Klein–Gordon–Schrödinger equations. We employ Fourier pseudo-spectral method for spatial discretization, resulting in a semi-discrete system. Subsequently, we employ the Petrov-Galerkin method in time to obtain a fully-discrete system. We rigorously demonstrate that the proposed scheme preserves the original energy of the target system. Furthermore, we have proved that the mass of the system is also approximately preserved. To assess the accuracy of our method, we provide a simple estimate of the local error, revealing that the proposed approach achieves a temporal order of 2s. Additionally, we extend the proposed methods to the damped Klein–Gordon–Schrödinger equations, and the corresponding fully-discrete scheme preserves the original energy dissipation law of the system. We present numerical examples to verify the accuracy and robustness of the proposed scheme.

Stochastic Dynamic Analysis of Multilayer Saturated Porous Cylindrical Structures Using the Meshless Local Petrov-Galerkin Method

Stochastic Dynamic Analysis of Multilayer Saturated Porous Cylindrical Structures Using the Meshless Local Petrov-Galerkin Method