Galerkin Spectral Method Research Articles

Non-polynomial Spectral-Galerkin Method for Time-Fractional Diffusion Equation on Unbounded Domain

Non-polynomial Spectral-Galerkin Method for Time-Fractional Diffusion Equation on Unbounded Domain

Efficient Stochastic Galerkin Spectral Methods for Optimal Control Problems Constrained by Fractional PDEs with Uncertain Inputs

This paper is devoted to designing fast solvers and efficient preconditioners for the optimal control problems (OCPs) constrained by stochastic fractional elliptic equations. We first prove the existence and uniqueness of the stochastic optimal control solution and then derive the stochastic optimality system. For the numerical approximation, we use the stochastic Galerkin spectral methods, which apply the stochastic Galerkin method for the discretization of random variables and employ the spectral-Galerkin approach for the approximation of spatial variables. To solve the large coupled saddle-point system resulted from discretization, we adopt the most commonly used MINRES method and a more effective PPCG method in low-rank matrix iteration format. Specially, we develop the efficient preconditioners based on the matrix decomposition method and the virtual variable method. We also study the eigenvalue distribution of the corresponding preconditioned matrix. Besides, for the approximation of the discretized state equation, we use the mean-based approximation and the Ullmann approximation to handle different values of the variance of the random inputs. Finally, we present numerical experiments to demonstrate the effectiveness of our solvers and preconditioners.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

Elasto-acoustic modeling and simulation for the seismic response of structures: The case of the Tahtalı dam in the 2020 [formula omitted]zmir earthquake

As a mean to assess the risk dam structures are exposed to during earthquakes, we employ an abstract mathematical, three dimensional, elasto-acoustic coupled wave-propagation model taking into account (i) the dam structure itself, embedded into (ii) its surrounding topography, (iii) different material soil layers, (iv) the seismic source as well as (v) the reservoir lake filled with water treated as an acoustic medium. As a case study for extensive numerical simulations we consider the magnitude 7 seismic event of the 30th of October 2020 taking place in the Icarian Sea (Greece) and the Tahtalı dam around 30km from there (Turkey). A challenging task is to resolve the multiple length scales that are present due to the huge differences in size between the dam building structure and the area of interest, considered for the propagation of the earthquake. Interfaces between structures and highly non-conforming meshes on different scales are resolved by means of a discontinuous Galerkin approach. The seismic source is modeled using inversion data about the real fault plane. Ultimately, we perform a real data driven, multi-scale, full source-to-site, physics based simulation based on the discontinuous Galerkin spectral element method, which allows to precisely validate the ground motion experienced along the Tahtalı dam, comparing the synthetic seismograms against actually observed ones. A comparison with a more classical computational method, using a plane wave with data from a deconvolved seismogram reading as an input, is discussed.

Regularity analysis for SVEEs with additive fBms and strong error estimates for the numerical approximations

Stochastic Volterra evolution equations with fractional Brownian motions have been widely used to describe many phenomena involving memories and fractional Gaussian noises in the real world. However, to our best knowledge, to date there is no rigorous mathematical analysis of the existence and uniqueness and regularity of the solutions and numerical approximations to such equations. This paper presents spatial and temporal Lp-regularity results for a class of stochastic Volterra evolution equations driven by fractional Brownian motions with the Hurst parameter H∈(0,12) and (12,1). The two cases are treated separately. It is mentioned that some clever techniques are used to overcome the difficulty of the singularity of the covariance of fractional Brownian motions with H∈(0,12). In addition, we give a full approximation for such equations, performed by the spectral Galerkin method in space and an implicit Euler method combined with a convolution quadrature in time. The obtained regularity properties of the equations help us to finish the strong convergence analysis of the fully discrete scheme. In final, several numerical experiments are demonstrated to substantiate the theoretical findings.

High-order discontinuous Galerkin approximation for a three-phase incompressible Navier–Stokes/Cahn–Hilliard model

In this work we introduce the development of a three-phase incompressible Navier–Stokes/Cahn–Hilliard numerical method to simulate three-phase flows, present in many industrial operations. The numerical method is then applied to successfully solve oil transport problems, such as those found in the oil and gas industry. The three-phase model adopted in this work is a Cahn–Hilliard diffuse interface model, which was derived by Boyer and Lapuerta (2006). The Cahn–Hilliard model is coupled to the kinetic-energy stable incompressible Navier–Stokes equations model derived by Manzanero et al. (2020). The spatial discretization uses a high-order discontinuous Galerkin spectral element method which yields highly accurate results in arbitrary geometries. An implicit–explicit (IMEX) method is adopted as temporal scheme for the Cahn–Hilliard equation, while Runge–Kutta 3 (RK3) is used for the Navier–Stokes equations. The developed numerical tool is validated with a manufactured solution test case and used to simulate multiphase flows in pipes, including and a three-phase T-shaped pipe intersection.

A spectral element method for modelling streamer discharges in low-temperature atmospheric-pressure plasmas

Streamers are ionization fronts that occur in gases at atmospheric and sub-atmospheric pressures. Numerical studies of streamers are important for practical applications but are challenging due to the multiscale nature of this discharge type. This paper introduces a spectral element method for modelling streamer discharges. The method is developed for Cartesian grids but can be extended to be used on unstructured meshes. The streamer model is based on the Poisson equation for the electric potential and the electron continuity equation. The Poisson equation is discretized via a spectral method based on the integral representation of the solution. The hierarchical Poincaré-Steklov (HPS) scheme is used to solve the resulting set of equations. The electron continuity equation is solved by means of the discontinuous Galerkin spectral element method (DGSEM). The DGSEM is extended by an alternative definition of the diffusion flux. A subcell finite volume method is used to stabilize the DGSEM scheme, if required. The entire simulation scheme is validated by solving a number of test problems and reproducing the results of previous studies. Adaptive mesh refinement is used to reduce the number of unknowns. The proposed method is found to be sufficiently fast for being used in practical applications. The flexibility of the method provides an interesting opportunity to broaden the range of problems that can be addressed in numerical studies of low-temperature plasma discharges.

Implicit two-derivative deferred correction time discretization for the discontinuous Galerkin method

In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method is high order accurate in space and time. As the novel scheme handles two time derivatives, the spatial operator for both derivatives has to be defined. This results in an extended system matrix of the scheme. We analyze this matrix regarding possible simplifications and an efficient way to solve the arising (non-)linear system of equations. It is shown how a carefully designed preconditioner and a matrix-free approach allow for an efficient implementation and application of the novel scheme. For both, linear advection and the compressible Euler equations, up to eighth order of accuracy in time is shown. Finally, it is illustrated how the method can be used to approximate solutions to the compressible Navier-Stokes equations.

A Fast Galerkin Approach for Solving the Fractional Rayleigh–Stokes Problem via Sixth-Kind Chebyshev Polynomials

Herein, a spectral Galerkin method for solving the fractional Rayleigh–Stokes problem involving a nonlinear source term is analyzed. Two kinds of basis functions that are related to the shifted sixth-kind Chebyshev polynomials are selected and utilized in the numerical treatment of the problem. Some specific integer and fractional derivative formulas are used to introduce our proposed numerical algorithm. Moreover, the stability and convergence accuracy are derived in detail. As a final validation of our theoretical results, we present a few numerical examples.

Optimized explicit Runge–Kutta schemes for high-order collocated discontinuous Galerkin methods for compressible fluid dynamics

In compressible computational fluid dynamics, the step size of explicit time integration schemes is often constrained by stability when high-order accurate spatial discretizations are used. We report a set of new optimized explicit Runge–Kutta schemes for the integration of systems of ordinary differential equations arising from the spatial discretization of wave propagation problems with high-order entropy stable collocated discontinuous Galerkin methods. The eigenvalues of the discrete spatial operator for the advection equation and the propagation of an isentropic vortex with the compressible Euler equations for various values of the problems' parameters are used to optimize the stability region of the proposed time integration schemes. To demonstrate the efficiency and the robustness of the methods, we solve the compressible turbulent flow past the Valeo controlled-diffusion airfoil and a delta wing at a Reynolds number of 8.3×105 and 106, respectively. A thorough analysis of the performance of the two families of optimized schemes revealed that methods generated using the spectra of the vortex problem are 6-to-20% faster than methods constructed using the spectra of the advection equation. Compared to widely used explicit Runge–Kutta schemes, the methods designed using the spectra of the vortex problem yield a time-to-solution saving of approximately 6-to-38%. For large-scale time-dependent partial differential equations computations, these gains mean saving hundreds of thousands if not millions of core hours. In addition, the new methods can be effectively and efficiently applied to integrate systems of ordinary differential equations arising from a wide range of spatial discretization, including discontinuous Galerkin spectral element methods, spectral difference methods, and flux reconstruction methods.

Spectral Galerkin treatment of linear one-dimensional telegraph type problem via the generalized Lucas polynomials

The major goal of this research is to develop and test a numerical technique for solving a linear one-dimensional telegraph problem. The generalized polynomials, namely, the generalized Lucas polynomials are selected as basis functions. To solve the linear one-dimensional telegraph type equation, we solve instead its corresponding integral equation via the application of the spectral Galerkin method that serves to convert the equation with its underlying conditions into a system of linear algebraic equations that may be solved by a suitable numerical solver. The convergence and error analysis of the generalized Lucas expansion are discussed in depth. The current analysis is based on the assumption that the problem’s solution is separable. Finally, some explanatory numerical examples are displayed together with comparisons to some other articles, to demonstrate the suggested method’s validity, applicability, and accuracy.

Local error estimate of L1 scheme for linearized time fractional KdV equation with weakly singular solutions

We consider the local error estimate of the L1 scheme on graded mesh for a linearized time fractional KdV equation with weakly singular solutions, where Legendre Petrov–Galerkin spectral method is used for the spatial discretization. Stability and convergence of the fully discrete scheme are rigorously established, and the pointwise-in-time error estimates are given to show that one can attain the optimal convergence order 2−α in positive time by mildly choosing the grading parameter r no more than 2 for all 0<α<1. Numerical results are presented to show that the error estimate is sharp.

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

SPECTRAL GALERKIN METHOD FOR STOCHASTIC SPACE-TIME FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

This work is devoted to deal with a stochastic space-time fractional integro-differential equation in the Hilbert space $ L^{2}(0,1) $, by studing its spatial approximation. Precisely, we use the spectral Galerkin method to prove that the spatial approximation converges strongly (i.e. in the space $ L^{p}(\Omega, L^{2}(0,1)) $), by imposing only a regularity condition on the initial value.

Dirichlet spectral-Galerkin approximation method for the simply supported vibrating plate eigenvalues

In this paper, we analyze and implement the Dirichlet spectral-Galerkin method for approximating simply supported vibrating plate eigenvalues with variable coefficients. This is a Galerkin approximation that uses the approximation space that is the span of finitely many Dirichlet eigenfunctions for the Laplacian. Convergence and error analysis for this method is presented for two and three dimensions. Here we will assume that the domain has either a smooth or Lipschitz boundary with no reentrant corners. An important component of the error analysis is Weyl’s law for the Dirichlet eigenvalues. Numerical examples for computing the simply supported vibrating plate eigenvalues for the unit disk and square are presented. In order to test the accuracy of the approximation, we compare the spectral-Galerkin method to the separation of variables for the unit disk. Whereas for the unit square we will numerically test the convergence rate for a variable coefficient problem.

A spectral method for stochastic fractional PDEs using dynamically-orthogonal/bi-orthogonal decomposition

Modeling uncertainty propagation in anomalous transport applications leads to formulating stochastic fractional partial differential equations (SFPDEs), which require special algorithms for obtaining satisfactory accuracy at reasonable computational complexity. Here, we consider a stochastic fractional diffusion-reaction equation and combine a Galerkin spectral method based on poly-fractonomials with the modal decomposition of the stochastic fields to formulate effective numerical methods for SFPDEs. Specifically, we employ a generalized Karhunen-Loève (KL) expansion and proper dynamically-orthogonal/bi-orthogonal (DO/BO) constraints to derive new Galerkin formulations for the mean solution, the time-dependent spatial basis, and the stochastic time-dependent coefficients. In addition, we employ a hybrid approach to tackle the singular limit of DO and BO equations for the case of deterministic initial conditions. The DO and BO formulations are mathematically equivalent but they exhibit computationally complementary properties. In demonstration examples, we investigate the interplay between randomness and non-locality and quantify this interaction for first time. In particular, we apply generalized polynomial chaos (gPC), DO, BO, and hybrid methods (i.e., combining gPC with DO or BO) to linear problems and (combining Quasi Monte Carlo (QMC) method with DO or BO) to nonlinear problems, and we compare our results to reference solutions obtained by well-resolved QMC simulations. We find that the DO and BO methods are both accurate approaches suitable for SFPDEs, with the BO method seemingly more accurate overall. The fractional order α affects strongly the accuracy of the solution and hence the required number of modes for multi-dimensional problems with parametric uncertainty. Both the DO and BO methods converge fast with respect to the number of modes, and they are especially effective for nonlinear problems and long-time integration for a modest number of stochastic dimensions. With regards to temporal discretization, the mean of the solution using the backward differentiation formula (BDF3) exhibits very high accuracy.

Large Deviations Principles for Symplectic Discretizations of Stochastic Linear Schrödinger Equation

In this paper, we consider the large deviations principles (LDPs) for the stochastic linear Schrödinger equation and its symplectic discretizations. These numerical discretizations are the spatial semi-discretization based on the spectral Galerkin method, and the further full discretizations with symplectic schemes in temporal direction. First, by means of the abstract Gärtner–Ellis theorem, we prove that the observable $B_{T}=\frac {u(T)}{T}$ , T > 0 of the exact solution u is exponentially tight and satisfies an LDP on $L^{2}(0, \pi ; \mathbb C)$ . Then, we present the LDPs for both $\{{B^{M}_{T}}\}_{T>0}$ of the spatial discretization $\{u^{M}\}_{M\in \mathbb N}$ and $\{{B^{M}_{N}}\}_{N\in \mathbb N}$ of the full discretization $\{{u^{M}_{N}}\}_{M,N\in \mathbb N}$ , where ${B^{M}_{T}}=\frac {u^{M}(T)}{T}$ and ${B^{M}_{N}}=\frac {{u^{M}_{N}}}{N\tau }$ are the discrete approximations of BT. Further, we show that both the semi-discretization $\{u^{M}\}_{M\in \mathbb N}$ and the full discretization $\{{u^{M}_{N}}\}_{M,N\in \mathbb N}$ based on temporal symplectic schemes can weakly asymptotically preserve the LDP of {BT}T> 0. These results show the ability of symplectic discretizations to preserve the LDP of the stochastic linear Schrödinger equation, and first provide an effective approach to approximating the large deviations rate function in infinite dimensional space based on the numerical discretizations.

Application of Galerkin spectral method for tearing mode instability

Magnetic reconnection and tearing mode instability play a critical role in many physical processes. The application of Galerkin spectral method for tearing mode instability in two-dimensional geometry is investigated in this paper. A resistive magnetohydrodynamic code is developed, by the Galerkin spectral method both in the periodic and aperiodic directions. Spectral schemes are provided for global modes and local modes. Mode structures, resistivity scaling, convergence and stability of tearing modes are discussed. The effectiveness of the code is demonstrated, and the computational results are compared with the results using Galerkin spectral method only in the periodic direction. The numerical results show that the code using Galerkin spectral method individually allows larger time step in global and local modes simulations, and has better convergence in global modes simulations.

Non-linear System of Multi-order Fractional Differential Equations: Theoretical Analysis and a Robust Fractional Galerkin Implementation.

This paper presents a comprehensive study of non-linear systems of multi-order fractional differential equations from aspects of theory and numerical approximation. In this regard, we first establish the well-posedness of the underlying problem by investigations concerning the existence, uniqueness, and influence of perturbed data on the behavior of the solutions as well as smoothness of the solutions under some assumptions on the given data. Next, from the numerical perspective, we develop and analyze a well-conditioned and high-order numerical approach based on an operational spectral Galerkin method. The main advantage of our strategy is that it characterizes the approximate solution via some recurrence formulas, instead of solving a complex non-linear block algebraic system that requires high computational costs. Moreover, the familiar spectral accuracy is justified in a weighted L^2-norm, and some practical test problems are provided to approve efficiency of the proposed method.

General linear and spectral Galerkin methods for the nonlinear two-sided space distributed-order diffusion equation

A high order numerical method is constructed for the two-dimensional nonlinear two-sided space distributed-order diffusion equation, in which the distributed-order integral is approximated by the Gauss-Legendre quadrature formula, the time and space are discretized by the general linear method and spectral Galerkin method, respectively. The proposed method is proved to be stable and convergent of order p in time, when the nonlinear term satisfies the local Lipschitz condition and the general linear method with generalized stage order p is coercive and algebraically stable. Moreover, the optimal error estimates in distributed-order integral and in space are also obtained. Finally, numerical experiments are presented to illustrate the theoretical estimates.