Galerkin Spectral Method Research Articles

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.

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new formulas concerned with these generalized polynomials are established. These generalized orthogonal polynomials are employed to treat the multi-term linear fractional differential equations (FDEs) that include some specific problems that arise in many applications. The basic idea behind the derivation of our proposed algorithm is built on utilizing a new power form representation of the shifted generalized Chebyshev polynomials along with the application of the spectral Galerkin method to transform the FDE governed by its initial conditions into a system of linear equations that can be efficiently solved via a suitable numerical solver. Some illustrative examples accompanied by comparisons with some other methods are presented to show that the presented algorithm is useful and effective.

Discontinuous Galerkin spectral element method for shock capturing with summation by parts properties

This paper presents a computational methodology developed for a high-order approximation of compressible fluid dynamics equations with discontinuities. The methodology is based on a discontinuous Galerkin spectral-element method (DGSEM) built upon a split discretization framework with summation-by-parts (SBP) property, which mimics the integration-by-parts operation in a discrete sense. To extend the split DGSEM framework to discontinuous cases, we implement a shock capturing method based on the entropy viscosity formulation. The developed high-order split-form DGSEM with shock-capturing methodology is subject to a series of evaluation on both one-dimensional and two-dimensional, continuous and discontinuous cases. Convergence of the method is demonstrated both for smooth and shocked cases that have analytical solutions. The 2D Riemann problem tests illustrate an accurate representation of all the relevant flow phenomena, such as shocks, contact discontinuities, and rarefaction waves. All test cases are able to run with a polynomial order of 7 or higher. The values of the tunable parameters related to the entropy viscosity are robust for both 1D and 2D test problems. We also show that higher-order approximations yield smaller errors than lower-order approximations, for the same number of total degrees of freedom.

Stability and Convergence of $L1$-Galerkin Spectral Methods for the Nonlinear Time Fractional Cable Equation

Stability and Convergence of $L1$-Galerkin Spectral Methods for the Nonlinear Time Fractional Cable Equation

A novel generalized symmetric spectral Galerkin numerical approach for solving fractional differential equations with singular kernel

<abstract><p>Polynomial based numerical techniques usually provide the best choice for approximating the solution of fractional differential equations (FDEs). The choice of the basis at which the solution is expanded might affect the results significantly. However, there is no general approach to determine which basis will perform better with a particular problem. The aim of this paper is to develop a novel generalized symmetric orthogonal basis which has not been discussed in the context of numerical analysis before to establish a general numerical treatment for the FDEs with a singular kernel. The operational matrix with four free parameters was derived for the left-sided Caputo fractional operator in order to transform the FDEs into the corresponding algebraic system with the aid of spectral Galerkin method. Several families of the existing polynomials can be obtained as a special case from the new basis beside other new families generated according to the value of the free parameters. Consequently, the operational matrix in terms of these families was derived as a special case from the generalized one up to a coefficient diagonal matrix. Furthermore, different properties relevant to the new generalized basis were derived and the error associated with function approximation by the new basis was performed based on the generalized Taylor's formula.</p></abstract>

Moment Preserving Fourier–Galerkin Spectral Methods and Application to the Boltzmann Equation

Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings.

Entropy-stable Gauss collocation methods for ideal magneto-hydrodynamics

In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms.Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, Chan et al. [1, “Efficient Entropy Stable Gauss Collocation Methods”. SIAM (2019)] presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. to the non-conservative GLM-MHD system.We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.

A fully-implicit hybridized discontinuous Galerkin spectral element method for two phase flow in petroleum reservoirs

A hybridized discontinuous Galerkin method with high order spectral element discretization is presented for modeling multiphase immiscible 2D fluid flow in petroleum reservoir simulations. The time dependent nonlinear governing equations for reservoir simulation are presented, and a fully coupled implicit time stepping procedure is implemented using a high order W-Rosenbrock algorithm. For incompressible fluids considered in the present study, the governing PDE's include an elliptic PDE for the pressure field and a parabolic convective diffusion equation for the saturation (water volume fraction) in a two phase oil-water system. The PDE systems are coupled through the highly nonlinear relative permeability and capillary pressure functions relating the fluid velocities of individual species to the individual species pressures and saturations. With HDG/spectral element (HDGSEM) spatial discretization of the coupled governing equations, the system constitutes an index 1 differential algebraic equation. It is shown that the W-Rosenbrock algorithm in combination with the HDGSEM method yields an efficient and effective solution procedure which demonstrates robust, high order temporal accuracy.Numerous physical tests cases are presented demonstrating that the versatility of this approach is sufficient to address the challenges in modeling heterogeneous permeability fields and geologic features of arbitrary shape with discontinuous permeability functions. It is also capable of resolving the sharp time dependent displacement fronts in oil-water systems encountered in secondary oil recovery. It is demonstrated that the high order polynomial basis functions (up to 12th order) used in the HDGSEM algorithm have the ability to resolve complex geometry and achieve accurate results utilizing a simple Cartesian mesh for non-conforming geological features. Eliminating the need to re-grid and conform the mesh to the boundaries of different geological features greatly simplifies the workflow for petroleum engineers.

A relaxation model for the non-isothermal Navier-Stokes-Korteweg equations in confined domains

The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals' theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Secondly, any real gas equation of state is based on a non-convex Helmholtz free energy potential which may cause the eigenvalues of the Jacobian of the first-order fluxes to become imaginary numbers inside the spinodal region.In this work, a thermodynamically consistent relaxation model is presented which is used to approximate the NSK equations. The model is complimented by thermodynamically consistent non-equilibrium boundary conditions which take contact angle effects into account. Due to the relaxation approach, the contribution of the Korteweg tensor in the linear momentum and total energy equations can be reduced to first-order terms which enables a straightforward implementation of contact angle boundary conditions in a numerical scheme. Moreover, the definition of a modified pressure function enables to formulate first-order fluxes which remain strictly hyperbolic in the entire spinodal region. The present work is a generalization of a previously presented parabolic relaxation model for the isothermal NSK equations.A high-order discontinuous Galerkin spectral element method which supports curved elements and hanging nodes is employed to discretize the system. The relaxation model and its corresponding boundary conditions are validated using solutions of the original NSK model and analytical results for one-, two- and three-dimensional test cases. The simulation of a spinodal decomposition in a three-dimensional porous structure underlines the capability of the presented approach.

A linear Galerkin numerical method for a quasilinear subdiffusion equation

We couple the L1 discretization for the Caputo derivative in time with the spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove the stability and convergence of the method with spectral accuracy in space. The temporal order depends on the regularity of the solution in time. Furthermore, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result, we find exact asymptotic values of the error constants along with their remainders for discretizations of the Caputo derivative and fractional integrals. These constants are the smallest possible, which improves previously established results from the literature.

Superconvergence of system of Volterra integral equations by spectral approximation method

In this article, we apply Jacobi spectral Galerkin, multi-Galerkin methods and their iterated versions to approximate the system of Volterra integral equations for smooth as well as weakly singular kernels and obtain the superconvergence results. Before doing this, first, we develop the regularity properties of the system of linear Volterra integral equations. We show that the Jacobi spectral iterated Galerkin approximation yields better converge rates over Jacobi spectral Galerkin method. We make the improvement of the superconvergence rates further for both smooth and weakly singular kernels in Jacobi spectral iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results.

A study of dual phage lag model of heat transfer for solidification of binary eutectic alloy

In this paper, we developed a generalized non-Fourier model of energy transfer for solidification of a binary eutectic system (Al-Cu alloy). The mathematical model is divided into three stages: stage I (liquid region), II (liquid and mushy region), and III (liquid, mushy and solid region). The analysis of the model has been done in dimensionless form. This model is a boundary value problem of the system of second-order hyperbolic partial differential equations for Stage I, the moving boundary problem of the System of second-order hyperbolic partial differential equations for Stage II and III. For the solution, we applied Laplace transform technique in Stage I and Legendre wavelet spectral Galerkin method in Stage II and III. The effect of unit-less parameters: Relaxation time, Predvoditelev numbers, dimensionless thermal conductivity, Fourier numbers, Stefan number, solid fraction on temperature, and position of moving interface are discussed in detail.

A spectral element solution of the two‐dimensional linearized potential flow radiation problem

AbstractWe present a scalable two‐dimensional Galerkin spectral element method solution to the linearized potential flow radiation problem for wave induced forcing of a floating offshore structure. The pseudo‐impulsive formulation of the problem is solved in the time domain using a Gaussian displacement signal tailored to the discrete resolution. The added mass and damping coefficients are then obtained via Fourier transformation. The spectral element method is used to discretize the spatial fluid domain, whereas the classical explicit 4‐stage fourth‐order Runge–Kutta scheme is employed for the temporal integration. Spectral convergence of the proposed model is established for both affine and curvilinear elements, and the computational effort is shown to scale with , with being the total number of grid points and . The solver is used to compute the hydrodynamic coefficients for several floating bodies and compared against known public benchmark results. The results show excellent agreement, ultimately validating the solver and emphasizing the geometrical flexibility and high accuracy and efficiency of the proposed solution strategy. Lastly, an extensive investigation of nonresolved energy from the pseudo‐impulse is carried out to characterize the induced spurious oscillations of the free surface quantities leading to a robust strategy for tuning the pseudo‐impulsive motion to the spatial discretization.

A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model

In this work, we propose an accurate, robust (the solution remains in the set of states), and stable discretization of a nonconservative model for the simulation of compressible multicomponent flows with shocks and material interfaces. We consider the gamma-based model by Shyue (1998) [58] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac (2019) [53] and Coquel et al. (2021) [15] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on a high-order discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points as proposed by Gassner (2013) [27] in the case of hyperbolic conservation laws. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization of general hyperbolic systems with conservative and nonconservative terms and derive the conditions to obtain a scheme that is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver for the gamma-based model that does not require a root-finding algorithm for the approximation of the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, to preserve uniform profiles of pressure and velocity across material interfaces (material interface preservation), and to satisfy a discrete minimum principle on the specific entropy and maximum principles on the parameters of the EOS. Third, the HLLC solver is applied at interfaces in the DGSEM, while we consider two kinds of fluctuations in the integrals over discretization elements: the former is entropy conservative (EC), while the latter preserves material interfaces (CP). Time integration is performed using high-order strong-stability preserving Runge-Kutta schemes. The fully discrete scheme is shown to preserve material interfaces with CP fluctuations. Under a given condition on the time step, both EC and CP fluctuations ensure that the cell-averaged solution remains in the set of states; satisfy a minimum principle on any convex entropy and maximum principles on the EOS parameters. These results have allowed us to use existing limiters in order to restore positivity, and discrete maximum principles of degrees-of-freedom within elements. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and accuracy of the proposed DGSEM with either CP, or EC fluctuations, while the scheme with CP fluctuations is shown to offer better resolution capabilities.

Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

Deep Adaptive Basis Galerkin Method for High-Dimensional Evolution Equations With Oscillatory Solutions

In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a deep adaptive basis Galerkin (DABG) method which employs the spectral-Galerkin method for time variable by a tensor-product basis for oscillatory solutions and the deep neural network method for high-dimensional space variables. The proposed method can lead to a linear system of differential equations having unknown DNNs that can be trained via the loss function. We establish a posteriori estimates of the solution error which is bounded by the minimal loss function and the term $O(N^{-m})$, where $N$ is the number of basis functions and $m$ characterizes the regularity of the equation, and show that if the true solution is a Barron-type function, the error bound converges to zero as $M=O(N^p)$ approaches to infinity where $M$ is the width of the used networks and $p$ is a positive constant. Numerical examples, including high-dimensional linear parabolic and hyperbolic equations, and a nonlinear Allen--Cahn equation are presented to demonstrate that the performance of the proposed DABG method is better than that of existing DNNs.

A high accurate and convergent numerical framework for solving high-order nonlinear Volterra integro-differential equations

This article discuss about the application of Jacobi spectral and pseudo-spectral Galerkin techniques for solving high-order nonlinear Volterra integro-differential equations. We supply strict error analysis for Jacobi spectral and pseudo-spectral Galerkin methods(JSGM,JPSGM) under reasonable assumptions, which indicate that the approximate solution errors decay exponentially in the sense of L∞ and Lω2 norm. Some numerical experiments are presented to verify the theoretical results. Superior numerical results with respect to some recent methods are provided in tables and figures.

An L1 type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion–reaction equations with fixed delay

A linearized spectral Galerkin/finite difference approach is developed for variable fractional-order nonlinear diffusion–reaction equations with a fixed time delay. The temporal discretization for the variable-order fractional derivative is performed by the L1-approximation. An appropriate basis function in terms of Legendre polynomials is used to construct the Galerkin spectral method for the spatial discretization of the second-order spatial operator. The main advantage of the proposed approach is that the implementation of the iterative process is avoided for the nonlinear term in the variable fractional-order problem. Convergence and stability estimates for the constructed scheme are proved theoretically by discrete energy estimates. Some numerical experiments are finally provided to demonstrate the efficiency and accuracy of the theoretical findings.

Subcell limiting strategies for discontinuous Galerkin spectral element methods

We present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations that allow for convex blending with the high-order variant with the goal of guaranteeing additional properties, such as bounds on physical quantities and/or guaranteed entropy dissipation. For an implementation of this main strategy, four main ingredients are identified that may be combined in a flexible manner: (i) a nodal high-order DG method on Legendre–Gauss–Lobatto nodes, (ii) a compatible robust subcell FV scheme, (iii) a convex combination strategy for the two schemes, which can be element-wise or subcell-wise, and (iv) a strategy to compute the convex blending factors, which can be either based on heuristic troubled-cell indicators, or using ideas from flux-corrected transport methods.By carefully designing the metric terms of the subcell FV method, the resulting methods can be used on unstructured curvilinear meshes, are locally conservative, can handle strong shocks efficiently while directly guaranteeing physical bounds on quantities such as density, pressure or entropy. We further show that it is possible to choose the four ingredients to recover existing methods such as a provably entropy dissipative subcell shock-capturing approach or a sparse invariant domain preserving approach.We test the versatility of the presented strategies and mix and match the four ingredients to solve challenging simulation setups, such as the KPP problem (a hyperbolic conservation law with non-convex flux function), turbulent and hypersonic Euler simulations, and MHD problems featuring shocks and turbulence.

Error Bounds of a Finite Difference/Spectral Method for the Generalized Time Fractional Cable Equation

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have min{2−α,2−β}-order convergence in time and spectral accuracy in space for smooth solutions, where α,β are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow.