Unified analysis for variational time discretizations of higher order and higher regularity applied to non-stiff ODEs

Time Discretization ◽

Quadrature Formulas ◽

Galerkin Methods ◽

Initial Value ◽

Discretization Methods ◽

Continuous Galerkin ◽

Higher Regularity ◽

AbstractWe present a unified analysis for a family of variational time discretization methods, including discontinuous Galerkin methods and continuous Galerkin–Petrov methods, applied to non-stiff initial value problems. Besides the well-definedness of the methods, the global error and superconvergence properties are analyzed under rather weak abstract assumptions which also allow considerations of a wide variety of quadrature formulas. Numerical experiments illustrate and support the theoretical results.

Variational time discretizations of higher order and higher regularity

BIT Numerical Mathematics ◽

10.1007/s10543-021-00851-6 ◽

2021 ◽

Author(s):

Simon Becher ◽

Gunar Matthies

Keyword(s):

Stability Problem ◽

Time Discretization ◽

Higher Order ◽

Collocation Methods ◽

Discretization Methods ◽

The Family ◽

Global Smoothness ◽

Continuous Galerkin ◽

Two Parameters ◽

Higher Regularity

AbstractWe consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied to Dahlquist’s stability problem, the presented methods provide the same stability properties as dG or cGP methods. Provided that suitable quadrature rules of Hermite type are used to evaluate the integrals in the variational conditions, the variational time discretization methods are connected to special collocation methods. For this case, we present error estimates, numerical experiments, and a computationally cheap postprocessing that allows to increase both the accuracy and the global smoothness by one order.

Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

Journal of Scientific Computing ◽

10.1007/s10915-021-01434-x ◽

2021 ◽

Vol 87 (1) ◽

Author(s):

Jan Nordström ◽

Andrew R. Winters

Keyword(s):

Discontinuous Galerkin ◽

Finite Difference Methods ◽

Basis Functions ◽

Galerkin Methods ◽

Theoretical Discussion ◽

Numerical Tests ◽

Dg Methods ◽

Theoretical Results ◽

Filtering Procedure

AbstractWe prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

Efficient High Order Semi-implicit Time Discretization and Local Discontinuous Galerkin Methods for Highly Nonlinear PDEs

Journal of Scientific Computing ◽

10.1007/s10915-016-0170-4 ◽

2016 ◽

Vol 68 (3) ◽

pp. 1029-1054 ◽

Cited By ~ 7

Author(s):

Ruihan Guo ◽

Francis Filbet ◽

Yan Xu

Keyword(s):

Discontinuous Galerkin ◽

Time Discretization ◽

High Order ◽

Nonlinear Pdes ◽

Implicit Time ◽

Galerkin Methods ◽

Local Discontinuous Galerkin ◽

Local Discontinuous Galerkin Methods ◽

Highly Nonlinear

A numerical dispersion-dissipation analysis of discontinuous Galerkin methods based on quadrilateral and triangular elements

Geophysics ◽

10.1190/geo2019-0109.1 ◽

2020 ◽

Vol 85 (3) ◽

pp. T101-T121 ◽

Cited By ~ 2

Author(s):

Xijun He ◽

Dinghui Yang ◽

Xueyuan Huang ◽

Xiao Ma

Keyword(s):

Discontinuous Galerkin ◽

Time Discretization ◽

Numerical Dispersion ◽

Numerical Dissipation ◽

Galerkin Methods ◽

Third Order ◽

Fully Discrete ◽

Triangular Elements ◽

Convergence Orders

The dispersive and dissipative properties of numerical methods are important for numerical modeling. We have evaluated a numerical dispersion-dissipation analysis for two discontinuous Galerkin methods (DGMs) — the flux-based DGM (FDGM) and the interior penalty DGM (IP DGM) for scalar wave equation. The semidiscrete analysis based on the plane-wave analysis is conducted for quadrilateral and triangular elements. Two kinds of triangular elements are taken into account. The fully discrete analysis for each method is conducted by incorporating a classic third-order total variation diminishing (TVD) Runge-Kutta (RK) time discretization. Our results indicate that FDGM produces smaller numerical dispersion than IP DGM, but it introduces more numerical dissipation. Notably, the two methods have different local convergence orders for numerical dispersion and dissipation. The anisotropy properties for different mesh types can also be identified. Several numerical experiments are carried out that verify some theoretical findings. The experiments exhibit that the numerical error introduced by FDGM is less than that introduced by IP DGM whereas the storage and calculation time of FDGM are greater than that of IP DGM. Overall, our work indicates that when both methods adopt third TVD RK time discretization, the computational efficiency of FDGM is slightly larger than IP DGM.

L∞-error estimates of discontinuous Galerkin methods with theta time discretization scheme for an evolutionary HJB equations

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.4306 ◽

2017 ◽

Vol 40 (12) ◽

pp. 4310-4319 ◽

Cited By ~ 2

Author(s):

Salah Boulaaras ◽

Mohamed Haiour ◽

Med Amine Bencheick Le hocine

Keyword(s):

Error Estimates ◽

Discontinuous Galerkin ◽

Time Discretization ◽

Galerkin Methods ◽

Discretization Scheme ◽

Hjb Equations

Discrete maximum principle for interior penalty discontinuous Galerkin methods

Open Mathematics ◽

10.2478/s11533-012-0154-z ◽

2013 ◽

Vol 11 (4) ◽

Cited By ~ 2

Author(s):

Tamás Horváth ◽

Miklós Mincsovics

Keyword(s):

Maximum Principle ◽

Discontinuous Galerkin ◽

Mesh Size ◽

Discrete Maximum Principle ◽

Galerkin Methods ◽

Penalty Parameter ◽

Discrete Level ◽

Interior Penalty ◽

AbstractA class of linear elliptic operators has an important qualitative property, the so-called maximum principle. In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator. We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter. We then investigate the sharpness of these conditions. The theoretical results are illustrated with numerical examples.

Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.200113.220213a ◽

2013 ◽

Vol 3 (1) ◽

pp. 59-80 ◽

Cited By ~ 36

Author(s):

Xinlong Feng ◽

Tao Tang ◽

Jiang Yang

Keyword(s):

Numerical Experiments ◽

Time Discretization ◽

Optimal Error Estimates ◽

Optimal Error ◽

Fully Discrete ◽

Phase Field Models ◽

Discretization Schemes ◽

Energy Stable ◽

Theoretical Results ◽

Crank Nicolson

AbstractIn this paper, stabilized Crank-Nicolson/Adams-Bashforth schemes are presented for the Allen-Cahn and Cahn-Hilliard equations. It is shown that the proposed time discretization schemes are either unconditionally energy stable, or conditionally energy stable under some reasonable stability conditions. Optimal error estimates for the semi-discrete schemes and fully-discrete schemes will be derived. Numerical experiments are carried out to demonstrate the theoretical results.

Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019080 ◽

2020 ◽

Vol 54 (2) ◽

pp. 705-726

Author(s):

Yong Liu ◽

Chi-Wang Shu ◽

Mengping Zhang

Keyword(s):

Error Estimates ◽

Discontinuous Galerkin ◽

Hyperbolic Equations ◽

Optimal Error Estimates ◽

Galerkin Methods ◽

Optimal Error ◽

Piecewise Polynomials ◽

Cartesian Meshes ◽

In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using Pk elements on uniform Cartesian meshes, and prove that the error in the L2 norm achieves optimal (k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.

Continuous stage stochastic Runge–Kutta methods

Advances in Difference Equations ◽

10.1186/s13662-021-03221-2 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Xuan Xin ◽

Wendi Qin ◽

Xiaohua Ding

Keyword(s):

Differential Equations ◽

Numerical Experiments ◽

Additive Noise ◽

Rooted Tree ◽

Order Theory ◽

Order Conditions ◽

Quadrature Formulas ◽

General Order ◽

Runge Kutta ◽

AbstractIn this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.