# Superconvergence analysis of local discontinuous Galerkin methods for linear convection–diffusion equations in one space dimension

- 35 Downloads

## Abstract

This paper is concerned with the superconvergence study of the local discontinuous Galerkin (LDG) method for one-dimensional time-dependent linear convection–diffusion equations, where the convection flux is taken as the upwind flux, while the diffusion fluxes chosen as the alternating fluxes. Superconvergence properties for both the solution itself and auxiliary variables are established. Precisely, we prove that, the LDG solutions are superconvergent with an order of \(k+2\) towards a particular projection of the exact solution and the auxiliary variable, and thus a \(k+1\)-th order superconvergence for the derivative approximation and a \(k+2\)-th order superconvergence for the function value approximation at a class of Radau points are obtained. Especially, we show that the convergence rate of the derivative approximation for the exact solution can reach \(k+2\) when the convection flux is the same as the diffusion flux, two order higher than the optimal convergence rate. Furthermore, a \(2k+1\)-th order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages, is also obtained under some suitable initial discretization. Numerical experiments indicate that most of our theoretical findings are optimal.

## Keywords

Superconvergence Local discontinuous Galerkin method Convection–diffusion equations Error estimates## Mathematics Subject Classification

26A27 65M15 42C10## Notes

## References

- Adjerid S, Massey TC (2006) Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput Methods Appl Mech Eng 195:3331–3346MathSciNetCrossRefGoogle Scholar
- Adjerid S, Weinhart T (2009) Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems. Comput Methods Appl Mech Eng 198:3113–3129MathSciNetCrossRefGoogle Scholar
- Adjerid S, Weinhart T (2011) Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems. Math Comput 80:1335–1367MathSciNetCrossRefGoogle Scholar
- Adjerid S, Devine KD, Flaherty JE, Krivodonova L (2002) A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput Methods Appl Mech Eng 191:1097–1112MathSciNetCrossRefGoogle Scholar
- Baker G, Dougalis VA, Karakashian OA (1983) Convergence of Galerkin approximations for Korteweg–de Vries equation. Math Comput 40:419–433MathSciNetCrossRefGoogle Scholar
- Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J Comput Phys 131:267–279MathSciNetCrossRefGoogle Scholar
- Cao W, Huang Q (2017) Superconvergence of local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J Sci Comput 72:761–791MathSciNetCrossRefGoogle Scholar
- Cao W, Zhang Z (2016) Superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. Math Comput 85:63–84MathSciNetCrossRefGoogle Scholar
- Cao W, Zhang Z, Zou Q (2014) Superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J Numer Anal 5:2555–2573MathSciNetCrossRefGoogle Scholar
- Cao W, Shu C-W, Yang Yang, Zhang Z (2015) Superconvergence of discontinuous Galerkin method for 2-D hyperbolic equations. SIAM J Numer Anal 53:1651–1671MathSciNetCrossRefGoogle Scholar
- Cao W, Li D, Yang Y, Zhang Z (2017) Superconvergence of discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations. ESAIM Math Model Numer Anal 51:467–486MathSciNetCrossRefGoogle Scholar
- Cao W, Shu C-W, Yang Y, Zhang Z (2018) Superconvergence of discontinuous Galerkin method for nonlinear hyperbolic equations. SIAM J Numer Anal 56:732–765MathSciNetCrossRefGoogle Scholar
- Castillo P (2003) A superconvergence result for discontinuous Galerkin methods applied to elliptic problems. Comput Methods Appl Mech Eng 192:4675–4685MathSciNetCrossRefGoogle Scholar
- Celiker F, Cockburn B (2007) Superconvergence of the numerical traces of discontinuous Galerkin and hybridized methods for convection–diffusion problems in one space dimension. Math Comput 76:67–96MathSciNetCrossRefGoogle Scholar
- Chen C, Hu S (2013) The highest order superconvergence for bi-\(k\) degree rectangular elements at nodes—a proof of \(2k\)-conjecture. Math Comput 82:1337–1355MathSciNetCrossRefGoogle Scholar
- Cheng Y, Shu C-W (2008) Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J Comput Phys 227:9612–9627MathSciNetCrossRefGoogle Scholar
- Cheng Y, Shu C-W (2010) Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension. SIAM J Numer Anal 47:4044–4072MathSciNetCrossRefGoogle Scholar
- Cockburn B, Shu C-W (1989) TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws, II: general framework. Math Comput 52:411–435MathSciNetzbMATHGoogle Scholar
- Cockburn B, Shu C-W (1998a) The Runge–Kutta discontinuous Galerkin method for conservation laws, V: multidimensional systems. J Comput Phys 141:199–224MathSciNetCrossRefGoogle Scholar
- Cockburn B, Shu C-W (1998b) The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J Numer Anal 35:2440–2463MathSciNetCrossRefGoogle Scholar
- Cockburn B, Lin S, Shu C-W (1989) TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws, III: one dimensioal systems. J Comput Phys 84:90–113MathSciNetCrossRefGoogle Scholar
- Cockburn B, Hou S, Shu C-W (1990) The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws, IV: the multidimensional case. Math Comput 54:545–581MathSciNetzbMATHGoogle Scholar
- Dong B, Shu C-W (2009) Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems. SIAM J Numer Anal 47:3240–3268MathSciNetCrossRefGoogle Scholar
- Gottlieb S, Shu C, Tadmor E (2001) Strong stability-preserving high-order time discretization methods. SIAM Rev 43:89–112MathSciNetCrossRefGoogle Scholar
- Guo L, Yang Y (2017) Superconvergence of discontinuous Galerkin methods for hyperbolic equations with singular initial data. Int J Numer Anal Model 14:342–354MathSciNetzbMATHGoogle Scholar
- Guo W, Zhong X, Qiu J (2013) Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach. J Comput Phys 235:458–485MathSciNetCrossRefGoogle Scholar
- Hufford C, Xing Y (2014) Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg–de Vries equation. J Comput Appl Math 255:441–455MathSciNetCrossRefGoogle Scholar
- Meng X, Shu C-W, Zhang Q, Wu B (2012a) Superconvergence of Discontinuous Galerkin method for scalar nonlinear conservation laws in one space dimension. SIAM J Numer Anal 50:2336–2356MathSciNetCrossRefGoogle Scholar
- Reed WH, Hill TR (1973) Triangular mesh for neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR-73-479, Los Alamos, NMGoogle Scholar
- Xia Y, Xu Y, Shu C-W (2007) Local discontinuous Galerkin methods for the Cahn–Hilliard type equations. J Comput Phys 227:472–491MathSciNetCrossRefGoogle Scholar
- Xu Y, Shu C-W (2004) Local discontinuous Galerkin methods for three classes of nonlinear wave equations. J Comput Math 22:250–274MathSciNetzbMATHGoogle Scholar
- Xu Y, Shu C-W (2012) Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations. SIAM J Numer Anal 50:79–104MathSciNetCrossRefGoogle Scholar
- Yan J, Shu C-W (2002) A local discontinuous Galerkin method for KdV type equations. SIAM J Numer Anal 40:769–791MathSciNetCrossRefGoogle Scholar
- Yang Y, Shu C-W (2012) Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J Numer Anal 50:3110–3133MathSciNetCrossRefGoogle Scholar
- Yang Y, Shu C-W (2015) Analysis of sharp superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. J Comput Math 33:323–340MathSciNetCrossRefGoogle Scholar