Galerkin Spectral Method Research Articles

CADET-Julia: Efficient and versatile, open-source simulator for batch chromatography in Julia

This study introduces CADET-Julia, an open-source, versatile and fast chromatography solver implemented in the Julia programming language. The software offers a platform for rapid prototyping and numerical refinement for a range of chromatography models, including the general rate model (GRM). The interstitial column mass balance was spatially discretized using a strong-form discontinuous Galerkin spectral element method (DGSEM) whereas a generalized spatial Galerkin spectral method (GSM) was applied for the particle mass balance. Three different benchmarks showcased the computational efficiency of CADET-Julia: A baseline benchmark was established by comparing the Julia implementation to a C++ implementation that employed the same mathematical methods and time integrator (CADET-DG). Various Julia time integrators were tested, and with the best-performing settings, the Julia implementation was benchmarked against CADET-DG and a finite volume (FV) based implementation in C++ (CADET-FV). Overall, Julia implementations performed better than C++ implementations and Galerkin methods were generally superior to finite volumes.

Energy bounds for discontinuous Galerkin spectral element approximations of well-posed overset grid problems for hyperbolic systems

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.

Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth‐Order Problems in Complex Regions

ABSTRACTIn this article, an efficient spectral Galerkin method, which is based on a mixed scheme, is proposed and studied for solving fourth‐order problems in complex regions. The fundamental idea behind this approach is to transform the initial problem into an equivalent form in cylindrical coordinates and to reshape the computational domain into a product‐type rectangular one, which facilitates the utilization of spectral methods. However, when considering the equivalent fourth‐order form directly in cylindrical coordinates, it introduces intricate pole conditions and variable coefficients, posing challenges to both theoretical analysis and algorithm implementation. To address this, we employ the orthogonality of Fourier series to further decompose it into a sequence of decoupled two‐dimensional fourth‐order eigenvalue problems. For each such problem, we introduce an auxiliary function to transform it into an equivalent second‐order coupled system. Building on this, we formulate a mixed variational formulation and discrete scheme, and prove the error estimates for eigenvalue and eigenfunction approximations. Furthermore, we extend this algorithm to the two‐dimensional complex domains. Finally, a series of numerical examples are presented, and the numerical results validate the effectiveness of the algorithm and the correctness of the theoretical results.

Superconvergence results for hypersingular integral equation of first kind by Chebyshev spectral projection methods

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O(N−2r), where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O(N−3r) and O(N−4r), by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results.

Spectral-Galerkin methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop two numerical methods, the Legendre-Galerkin method and the generalized Log orthogonal functions Galerkin method for numerically solving the fully nonlinear Monge-Ampère equation. Both methods are constructed based on the vanishing moment approach. To address both solution stability and computational efficiency, we propose a multiple-level framework for resolving discretization schemes. The mathematical justifications of the new approaches and the error estimates for the Legendre-Galerkin method are established. Numerical experiments validate the accuracy of our methods, and a comparative experiment demonstrates the advantage of Log orthogonal functions for problems with corner singularities. The results highlight that our methods have high-order accuracy and small computational cost.

Explicit exponential Runge–Kutta methods for semilinear time-fractional integro-differential equations

In this work, we consider and analyze explicit exponential Runge–Kutta methods for solving semilinear time-fractional integro-differential equation, which involves two nonlocal terms in time. Firstly, the temporal Runge–Kutta discretizations follow the idea of exponential integrators. Subsequently, we utilize the spectral Galerkin method to introduce a fully discrete scheme. Then, we mainly focus on discussing the one-stage and two-stage methods for solving the proposed semilinear problem. Based on special abstract settings, we perform the convergence analysis for the proposed two different stage methods. In this process, we heavily use estimates about the operator family {S̃(t)}, and in combination with Lipschitz continuous condition. Finally, some numerical experiments confirm theoretical results. Meanwhile, applying this scheme to the related linear problem yields high-order convergence, highlighting the advantages of explicit exponential Runge–Kutta methods.

Superconvergent method for weakly singular Fredholm-Hammerstein integral equations with non-smooth solutions and its application

In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results.

Energy-conserving SAV-Hermite–Galerkin spectral scheme with time adaptive method for coupled nonlinear Klein–Gordon system in multi-dimensional unbounded domains

In this article, we construct an efficient numerical scheme to solve the coupled nonlinear Klein–Gordon system in a multi-dimensional unbounded domain Rd(d=1,2,3). We first use the SAV method to equivalently deform the original system, then we treat the nonlinear part of the system explicitly in temporal discretization. Thanks to the SAV method, we ensure the unconditional energy conservation of the discrete system, as a cost, we only need to solve two completely linearized decoupled systems. To avoid the errors caused by domain truncation and the construction of artificial boundary conditions, we apply the Hermite–Galerkin spectral method with scaling factor for spatial discretization. In order to save computational costs, we utilize two time adaptive strategies to adjust the time step size, through which we can improve computational efficiency without sacrificing accuracy. Finally, we presented numerical experiments to demonstrate the accuracy, efficiency, and energy conservation properties of our algorithm, and applied it to simulate the interactions of two-dimensional and three-dimensional vector solitons.

Spectral Galerkin Methods for Riesz Space-Fractional Convection–Diffusion Equations

This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods.

Maximum principle preserving time implicit DGSEM for linear scalar hyperbolic conservation laws

The properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward Euler time stepping are investigated for the approximation of hyperbolic linear scalar conservation equation in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport (FCT) limiting [5] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow us to use a linear scaling limiter [58] in order to impose a maximum principle at nodal values within elements, while limiting the cell average with the FCT limiter improves the accuracy of the limited solution. Then, we investigate the inversion of the linear systems resulting from the time implicit discretization at each time step. We prove that the diagonal blocks are invertible and provide efficient algorithms for their inversion. Numerical experiments in one and two space dimensions are presented to illustrate the conclusions of the present analyses.

Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel

Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel

Convergence analysis of a fully discrete scheme for diffusion-wave equation forced by tempered fractional Brownian motion

In this paper, we investigate a fully discrete approximation of stochastic diffusion-wave equation driven by additive tempered fractional Gaussian noise. This additive noise exhibits semi-long range dependence. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative and a tempered fractional Brownian motion. The space discretization is achieved via the spectral Galerkin method. The Caputo fractional derivative of order α∈(1,2) is approximated by using the Grünwald–Letnikov formula. Combining Mittag-Leffler function, Laplace transform and z-transform, we establish the error estimates of the full discretization in the sense of mean-squared L2-norm. Numerical experiments are presented to confirm the strong convergence rates.

A parametric study of precession driven dynamos inside a sphere

The dynamo actions of an electrically conducting fluid in a precessing sphere are investigated over a wide range of parameters by direct numerical simulation using a Galerkin spectral method. The focus of this work is to identify the most promising parameter regimes for the dynamo action and to investigate the characteristics of the magnetic field generated by precession. The influence of different nutation angles (30°,60°,90°) and different precession ratios on the ability to drive dynamo action are investigated. The optimal angle for dynamo actions is found at 90°, followed by 60° with retrograde precession. A moderate precession ratio around 0.3 is shown to be more feasible for dynamo actions. A rich set of self-sustained dynamo solutions are obtained in the parameter space we explored, including steady, periodic, quasi-periodic, and turbulent dynamos. The structure of the generated magnetic fields is analyzed by using helical wave decomposition. None of the precession driven dynamos we obtained produce a predominantly dipolar field, contrary to the convection driven dynamos. The long-time evolution of the magnetic dipole moment is investigated and different types of polarity reversals are observed.

Legendre Galerkin spectral collocation least squares method for the Darcy flow in homogeneous medium and non-homogeneous medium

The Darcy's equation consists of the mass conservation equation and the Darcy's law that involves the hydraulic potential (or called pressure) and the fluid velocity, which governs the flow of an incompressible fluid through a porous medium. In this paper, we investigate the Legendre Galerkin spectral collocation least squares method for approximating the problem of Darcy flow in homogeneous medium and non-homogeneous medium, respectively. The proposed scheme can be solved the approximate solutions of the hydraulic potential and the average velocity of the fluid simultaneously. A symmetric positive definite coefficient matrix of the corresponding linear algebra equation is obtained by applying our scheme. Numerical examples are presented to validate the efficiency and accuracy of the proposed scheme.

Effective numerical simulation of time fractional KdV equation with weakly singular solutions

A second-order time-stepping method for numerically solving the linearized time fractional KdV equation (TFKDVE) whose solution has initial singularity is investigated. Alikhanov’s scheme is used to discretize the Caputo fractional derivative on temporal graded mesh, and the space is approximated by a Petrov–Galerkin spectral method. Detailed stability and error estimate of the full-discrete scheme are given, and the final error bound is [Formula: see text]-robust. Numerical examples are given to illustrate the sharpness of the error analysis.

Error analysis of a fully discrete method for time-fractional diffusion equations with a tempered fractional Gaussian noise

We develop and analyze a fully discrete method, in order to solve numerically time-fractional diffusion equations driven by additive tempered fractional Gaussian noise. The stochastic model involves a Caputo fractional derivative of order α∈(0,1). We discuss the regularity results of the solution by using the inversion of Laplace transform, the Wright function and covariance function of stochastic process. A spectral Galerkin method is applied to approximate the stochastic model in space. Time discretization is achieved by using firstly a Riemann–Liouville derivative to reformulate the spatial semi-discrete form, and then applying the Grünwald–Letnikov formula. We derive the error estimates of the discrete methods, based on regularity results of the solution. Finally, extensive numerical experiments are presented to confirm our theoretical analysis.

A Verification Suite of Test Cases for the Barotropic Solver of Ocean Models

AbstractThe development of any atmosphere or ocean model warrants a suite of test cases (TCs) to verify its spatial and temporal discretizations, order of accuracy, stability, reproducibility, portability, scalability, etc. In this paper, we present a suite of shallow water TCs designed to verify the barotropic solver of atmosphere and ocean models. These include the non‐dispersive coastal Kelvin wave; the dispersive inertia‐gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non‐linear manufactured solution. These TCs check the implementation of the linear pressure gradient term; the linear constant or variable‐coefficient Coriolis and bathymetry terms; and the non‐linear advection terms. Simulation results are presented for a variety of time‐stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high‐order discontinuous Galerkin spectral element method. The experimental procedure for conducting these numerical experiments is detailed. It underscores several key considerations that vary depending on the chosen spatial discretization method. Finally, convergence studies of every TC are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions.

A linearly implicit energy-stable scheme for critical dissipative surface quasi-geostrophic flows

In this paper, we propose an effective linearly implicit unconditional energy-stable scheme for surface quasi-geostrophic flows based on the scalar auxiliary variable approach and the Fourier spectral Galerkin method. Compared with traditional numerical methods, our scheme has constant coefficient matrices at each time step, and the numerical solutions are consistent with the dissipation laws for modified energy. By treating linear terms implicitly and nonlinear terms explicitly, we derive the dissipation laws for discrete modified surface kinetic energy and Hamiltonian. To reduce the aliasing error induced by the Fourier spectral Galerkin method, we implement a 2/3 de-aliasing technique for the nonlinear terms. Furthermore, the integration concerning energy in our numerical scheme is exact due to the Fourier spectral Galerkin method. Numerical experiments are presented to verify the stability and efficiency of the proposed scheme.

A space-time Galerkin Müntz spectral method for the time fractional Fokker–Planck equation

In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results.

Accuracy assessment of discontinuous Galerkin spectral element method in simulating supersonic free jets

Accuracy assessment of discontinuous Galerkin spectral element method in simulating supersonic free jets