Galerkin Method Research Articles

Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations

This investigation discusses a numerical approach to solving the hyperbolic telegraph equation in both one and two dimensions. Applying the Galerkin method is the basis of this approach. We use appropriate combinations of third-kind modified shifted Chebyshev polynomials (3KMSCPs) as basis functions to transform the governing partial differential equations into a collection of algebraic equations. Through spectral Galerkin techniques, we establish the convergence error to demonstrate that our algorithm is more effective and efficient. Five examples are examined to verify the effectiveness and resilience of the applied method by comparing errors and illustrating the results. Our results show that the current numerical solutions align closely with exact solutions. The current algorithm is simple to set up and is better suited to solving certain difficult partial differential equations.

The ultra-weak discontinuous Galerkin method for time-fractional Burgers equation

The ultra-weak discontinuous Galerkin method for time-fractional Burgers equation

A Stokes–Dual–Porosity–Poroelasticity Model and Discontinuous Galerkin Method for the Coupled Free Flow and Dual Porosity Poroelastic Medium Problem

A Stokes–Dual–Porosity–Poroelasticity Model and Discontinuous Galerkin Method for the Coupled Free Flow and Dual Porosity Poroelastic Medium Problem

Stability and bifurcation analysis of simply supported rectangular nanoplate embedded on visco-Pasternak foundation based on first-order shear deformation theory

Vibrations of simply supported nanoplates embedded on visco-Pasternak foundation and stressed by in-plane periodic forces are investigated. The governing size-dependent equations employ the first-order shear deformation theory, nonlinear von Kármán strains, and the nonlocal elasticity theory. Reducing the governing system is carried out by the Bubnov-Galerkin method, which is based on two-mode model, thus the resulting system contains two ordinary differential equations with size- and time-dependent coefficients. Analysing the linearized system, the stability of the nanoplate structure is studied depending on the foundation parameters, force parameters and small-scale parameter. Nonlinear dynamics approaches are employed to determine the nature of vibration after the loss of stability. Time history, Poincare section, Lyapunov exponent are analysed for chosen values of excitation parameters, visco-Pasternak foundation parameters to indicate small-scale effects and nonlinear effects. The novelty of the work consists of the combined application of the nonlocal theory, first-order shear deformation theory, Floquet theory, methods of nonlinear dynamics and obtaining new results by a numerical experiment.

Energy-conserving Particle-In-Cell scheme based on Galerkin methods with sparse grids

Energy-conserving Particle-In-Cell scheme based on Galerkin methods with sparse grids

Vibration suppression analysis of iced transmission lines under axial time-delay velocity feedback strategy

To ensure the safety of power energy transmission channel and mitigate the harm caused by galloping of iced transmission lines, the axial time-delay velocity feedback strategy is adopted to suppress the galloping. The partial differential equation of galloping with axial time-delay velocity feedback strategy is established based on the variational principle for Hamiltonian. Then, the partial differential equation of galloping is transformed into ordinary differential equation based on normalization and the Galerkin method. The primary amplitude-frequency response equation, the first-order steady-state approximate solution, and the harmonic amplitude-frequency response equation are derived by the multiscale method. The impact of different parameters such as time-delay value, control coefficient, and amplitude of external excitation on the galloping response are analyzed. The amplitude under the primary resonance exhibits periodicity as time-delay value varies. The amplitude diminishes with increased control coefficient and increases with external excitation. Comprehensive consideration of various influences of parameters on vibration characteristics is crucial when employing the axial time-delay velocity feedback strategy to suppress galloping. Therefore, to achieve the best vibration suppression effect, it is crucial to adjust the time-delay parameter for modifying the range and amplitude of the resonance zone. The conclusions obtained by this study are expected to advance the refinement of active control techniques for iced transmission lines, and may provide valuable insights for practical engineering applications.

Corrigendum: Discontinuous Galerkin methods for the Vlasov-Poisson system

Corrigendum: Discontinuous Galerkin methods for the Vlasov-Poisson system

Discontinuous Galerkin Methods for Auto-Convolution Volterra Integral Equations

Discontinuous Galerkin Methods for Auto-Convolution Volterra Integral Equations

Numerical solutions of multi-term fractional reaction-diffusion equations

<p>In this paper, we have proposed a numerical approach based on generalized alternating numerical fluxes to solve the multi-term fractional reaction-diffusion equation. This type of equation frequently arises in the mathematical modeling of ultra-slow diffusion phenomena observed in various physical problems. These phenomena are characterized by solutions that exhibit logarithmic decay as time $ t $ approaches infinity. For spatial discretization, we employed the discontinuous Galerkin method with generalized alternating numerical fluxes. Temporal discretization was handled using the finite difference method. To ensure the robustness of the proposed scheme, we rigorously established its unconditional stability through mathematical induction. Finally, we conducted a series of comprehensive numerical experiments to validate the accuracy and efficiency of the scheme, demonstrating its potential for practical applications.</p>

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

The primary resonance of a Duffing oscillator with a restoring force of fractional-order derivatives by the extended Galerkin method

The primary resonance of a Duffing oscillator with a restoring force of fractional-order derivatives by the extended Galerkin method

An Energy Stable Local Discontinuous Galerkin Method for a Binary Compressible Flow

An Energy Stable Local Discontinuous Galerkin Method for a Binary Compressible Flow

Structure-preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system

Structure-preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system

Review of research on the vibration and buckling of functionally graded spherical shells

Spherical shells are important components in many aerospace and engineering structures. A functionally graded material (FGM) spherical shell can have its material distribution varied with the change of the radius direction, which can enhance the bearing capacity and the stability of the thin-walled shells. However, due to the special geometric characteristics of the spherical shell, the simplicity of its geometry will lead to great complexity in its vibration and buckling analysis processes. This paper mainly reviews the previous analytical and numerical research on the vibration and buckling of the FGM spherical shell. The shell's middle plane equilibrium equations are derived from plate theory. The Galerkin method, the Ritz method, the power series solution method, and other methods are introduced to carry out the vibration and buckling analysis. The frequencies, mode shapes, and the critical buckling load of the FGM spherical shells, which are under various boundary conditions and external pressure.

Synchronous detection of multiple micro-substances with a single output channel using a weakly coupled resonator array

Abstract In this paper, a multiple micro-substances sensing scheme with a single output channel using a weakly coupled resonator array is proposed. An analytical model of the resonator using the Euler-Bernoulli beam theory has been developed, as well as the dynamic behavior has been further explored using the Galerkin method. We have discovered numerically that, in the weakly coupled cantilever array, the A-f (amplitude-frequency) curve of any cantilever physically reflects the vibration features of other cantilevers. Hence, by measuring the output signal of a single cantilever, multiple substances applied on each cantilever respectively can be identified and detected synchronously. A single output channel via a weakly coupled micro-resonator array is constructed and validated numerically for picogram level mass detection. Equivalent experiments with a macro coupled five-cantilever array have been further conducted for verification. Under a common sweep-driven signal, the five analytes applied on each cantilever with masses at microgram level can be detected synchronously and independently by measuring the frequency shifts of the five resonant peaks of the center cantilever. Multiple substances sensing with a single output channel and a single driving signal is thus realized with low relative errors and high linearities. By enhancing the driving voltage, the mass resolution can also be improved via Duffing bifurcation. This work not only reveals a new coupled vibration behavior but also provides a new avenue for multiple analytes detection.

Discontinuous Galerkin discretization of coupled poroelasticity–elasticity problems

Abstract This work is concerned with the analysis of a space–time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic–elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is coupled with a dG time integration scheme, resulting in a full space–time dG discretization. We present the stability analysis for both semidiscrete and fully discrete formulations, and derive error estimates in suitable energy norms. The method is applied to various numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.

A Non‐Dissipative, Energy‐Conserving, Arbitrary High‐Order Numerical Method and Its Efficient Implementation for Incompressible Flow Simulation in Complex Geometries

ABSTRACTIn the inviscid limit, the energy of a velocity field satisfying the incompressible Navier–Stokes equations is conserved. Non‐dissipative numerical methods that discretely mimic this energy conservation feature have been demonstrated in the literature to be extremely valuable for robust and accurate large‐eddy simulations of high Reynolds number incompressible turbulent flows. For complex geometries, such numerical methods have been traditionally developed using the finite volume framework and they have been at best second‐order accurate. This paper proposes a non‐dissipative and energy‐conserving numerical method that is arbitrary high‐order accurate for triangle/tetrahedral meshes along with its efficient implementation. The proposed method is a Hybridizable Discontinuous Galerkin (HDG) method. The crucial ingredients of the numerical method that lead to the discretely non‐dissipative and energy‐conserving features are: (i) The tangential velocity on the interior faces, just for the convective term, is set using the non‐dissipative central scheme and the normal velocity is enforced to be continuous, that is, (div)‐conforming. (ii) An exactly (pointwise) divergence‐free basis is used in each element of the mesh for the stability of the convective discretization. (iii) The combination of velocity, pressure, and velocity gradient spaces is carefully chosen to avoid using stabilization which would introduce numerical dissipation. The implementation description details our choice of the orthonormal and degree‐ordered basis for each quantity and the efficient local and global problem solution using them. Numerical experiments demonstrating the various features of the proposed method are presented. The features of this HDG method make it ideal for high‐order LES of incompressible flows in complex geometries.

DISSIPATION OF WAVE ENERGY AND MITIGATION OF WAVE FORCE BY MULTIPLE FLEXIBLE POROUS PLATES

Abstract The hydroelastic interaction between water waves and multiple submerged porous elastic plates of arbitrary lengths in deep water is examined using the Galerkin approximation technique. We observe the influence of flexible porous plates of arbitrary lengths by analysing the reflection coefficient, dissipated energy and wave forces acting on the plates. Results are presented for various values of angle of incidence, separation lengths of plates, porosity levels, submergence depth and flexural rigidity. The convergence and accuracy of the method are verified by comparing the results with existing literature. The significant impact of flexural rigidity in the presence of porosity on wave reflection, dissipated energy and wave forces is demonstrated. Moreover, a notable reduction in wave load is observed with an increase in the number of plates.

Discontinuous Galerkin method with constrained transport for modelling solar wind in spherical coordinates

Abstract In this paper, the discontinuous Galerkin (DG) method is used to simulate the steady-state solar wind with constrained transport method. A cell-based piecewise quadratic polynomial solution is obtained through the discontinuous Galerkin scheme, while the face-averaged magnetic field is discretized using constrained-transport method. Then, a globally divergence-free magnetic field is constructed by combining the quadratic polynomial of magnetic field with the face-averaged magnetic field. For the time discretization, third-order Runge-Kutta method is applied. The resulting method can reach three order of accuracy in space and time. We test the method by computing the solar wind in Carrington rotation 2234, and compare the result with both remote sensing observations from SOHO and SDO as well as in-situ observations from Parker Solar Probe (PSP) and OMNI. The agreement between the modeled results and the observations demonstrates the capability of the developed scheme.

The simulation of spatio-temporal neural field equations with delay depending on the position of neural fibers using the Galerkin method based on moving least squares

The simulation of spatio-temporal neural field equations with delay depending on the position of neural fibers using the Galerkin method based on moving least squares