High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation
This paper deals with a high-order accurate Runge
Kutta Discontinuous Galerkin (RKDG) method for the numerical
solution of the wave equation, which is one of the simple case of a
linear hyperbolic partial differential equation. Nodal DG method is
used for a finite element space discretization in 'x' by discontinuous
approximations. This method combines mainly two key ideas which
are based on the finite volume and finite element methods. The
physics of wave propagation being accounted for by means of
Riemann problems and accuracy is obtained by means of high-order
polynomial approximations within the elements. High order accurate
Low Storage Explicit Runge Kutta (LSERK) method is used for
temporal discretization in 't' that allows the method to be nonlinearly
stable regardless of its accuracy. The resulting RKDG
methods are stable and high-order accurate. The L1 ,L2 and L∞ error
norm analysis shows that the scheme is highly accurate and effective.
Hence, the method is well suited to achieve high order accurate
solution for the scalar wave equation and other hyperbolic equations.
[1] W.H. Reed and T. Hill. Triangular mesh methods for the neutron
transport equation. Tech. report LA-UR-73-479, Los Alamos Scientific
Laboratory, 1973.
[2] P. LeSaint and P. A. Raviart, On a finite element method for solving the
neutron transport equation, Mathematical aspects of finite elements in
partial differential equations (C. de Boor, Ed.), Academic Press, 89
(1974).
[3] C. Johnson and J. PitkËôranta, An analysis of the discontinuous Galerkin
method for a scalar hyperbolic equation, Math. Comp. 46, 1 (1986).
[4] G. R. Richter, An optimal-order error estimate for the discontinuous
Galerkin method, Math. Comp. 50, 75(1988).
[5] T. Peterson, A note on the convergence of the discontinuous Galerkin
method for a scalar hyperbolic equation, SIAM J. Numerical. Analysis.
28, 133 (1991).
[6] B. Cockburn and C. W. Shu, "The Runge-Kutta Discontinuous Galerkin
Method for conservation laws V: Multidimensional System," Journal of
Computational Physics, Vol. 141, pp. 199-224, 1998.
[7] B. Cockburn and C.-W. Shu. The TVB Runge-Kutta local projection
discontinuous Galerkin finite element method for conservation laws V:
multidimensional systems. J. Comput. Phys., 141:199-224, 1998.
[8] B. Cockburn, S. Hou and C.-W. Shu. The TVB Runge-Kutta local
projection discontinuous Galerkin finite element method for
conservation laws IV: the multidimensional case. Math. Comp., 54:545-
581, 1990.
[9] B. Cockburn, S.-Y. Lin and C.-W. Shu. TVB Runge-Kutta local
projection discontinuous Galerkin finite element method for
conservation laws III: one dimensional systems. J. Comput. Phys.,
52:411-435, 1989.
[10] B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection
discontinuous Galerkin finite element method for conservation laws II:
general framework. Math. Comp., 52:411- 435, 1989.
[11] C.-W. Shu. TVB uniformly high-order schemes for conservation laws.
Math. Comp.,49:105-121, 1987.
[12] C.-W. Shu and S. Osher. Efficient Implementation of essentially nonoscillatory
shock capturing schemes. J. Comput. Phys., 77:439-471,
1988.
[13] ZhiliangXu, Yingjie Liu "A Conservation Constrained Runge-Kutta
Discontinuous Galerkin Method with the Improved CFL Condition for
Conservation Laws" February 16, 2010.
[14] H. Luo, J. D. Baum, and R. Lohner, "A Hermite WENO-based Limiter
for Discontinuous Galerkin Method on Unstructured Grids", 45th AIAA
Aerospace Sciences Meeting and Exhibit, 8-11, January 2007, Reno,
Nevada
[15] Jan S. Hesthaven Tim Warburton "Nodal Discontinuous Galerkin
Methods"- Algorithms, Analysis, and Applications" (2008).
[16] B. Cockburn and C.WShu "Runge-Kutta Discontinuous Galerkin
Methods for Convection-Dominated Problems" Journal of Scientific
Computing, Vol. 16, No. 3, September 2001 (2001).
[17] B. Cockburn, G.E. Karniadakis and C.-W. Shu. (2000).The development
of discontinuous Galerkin methods. In B. Cockburn, G.E. Karniadakis
and C.-W. Shu (eds), Discontinuous Galerkin Methods. Theory,
Computation and Application, Lecture Notes in Computational Science
and Engineering, Vol. 11, Springer-Verlag, pp. 3-50.
[18] Fareed Ahmed, Faheem Ahmed, Yongheng Guo, andYong Yang, "A
High Order Accurate Nodal Discontinuous Galerkin Method (DGM) for
Numerical Solution of Hyperbolic Equation," International Journal of
Applied Physics and Mathematics vol. 2, no. 5, pp. 362-364, 2012.
[1] W.H. Reed and T. Hill. Triangular mesh methods for the neutron
transport equation. Tech. report LA-UR-73-479, Los Alamos Scientific
Laboratory, 1973.
[2] P. LeSaint and P. A. Raviart, On a finite element method for solving the
neutron transport equation, Mathematical aspects of finite elements in
partial differential equations (C. de Boor, Ed.), Academic Press, 89
(1974).
[3] C. Johnson and J. PitkËôranta, An analysis of the discontinuous Galerkin
method for a scalar hyperbolic equation, Math. Comp. 46, 1 (1986).
[4] G. R. Richter, An optimal-order error estimate for the discontinuous
Galerkin method, Math. Comp. 50, 75(1988).
[5] T. Peterson, A note on the convergence of the discontinuous Galerkin
method for a scalar hyperbolic equation, SIAM J. Numerical. Analysis.
28, 133 (1991).
[6] B. Cockburn and C. W. Shu, "The Runge-Kutta Discontinuous Galerkin
Method for conservation laws V: Multidimensional System," Journal of
Computational Physics, Vol. 141, pp. 199-224, 1998.
[7] B. Cockburn and C.-W. Shu. The TVB Runge-Kutta local projection
discontinuous Galerkin finite element method for conservation laws V:
multidimensional systems. J. Comput. Phys., 141:199-224, 1998.
[8] B. Cockburn, S. Hou and C.-W. Shu. The TVB Runge-Kutta local
projection discontinuous Galerkin finite element method for
conservation laws IV: the multidimensional case. Math. Comp., 54:545-
581, 1990.
[9] B. Cockburn, S.-Y. Lin and C.-W. Shu. TVB Runge-Kutta local
projection discontinuous Galerkin finite element method for
conservation laws III: one dimensional systems. J. Comput. Phys.,
52:411-435, 1989.
[10] B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection
discontinuous Galerkin finite element method for conservation laws II:
general framework. Math. Comp., 52:411- 435, 1989.
[11] C.-W. Shu. TVB uniformly high-order schemes for conservation laws.
Math. Comp.,49:105-121, 1987.
[12] C.-W. Shu and S. Osher. Efficient Implementation of essentially nonoscillatory
shock capturing schemes. J. Comput. Phys., 77:439-471,
1988.
[13] ZhiliangXu, Yingjie Liu "A Conservation Constrained Runge-Kutta
Discontinuous Galerkin Method with the Improved CFL Condition for
Conservation Laws" February 16, 2010.
[14] H. Luo, J. D. Baum, and R. Lohner, "A Hermite WENO-based Limiter
for Discontinuous Galerkin Method on Unstructured Grids", 45th AIAA
Aerospace Sciences Meeting and Exhibit, 8-11, January 2007, Reno,
Nevada
[15] Jan S. Hesthaven Tim Warburton "Nodal Discontinuous Galerkin
Methods"- Algorithms, Analysis, and Applications" (2008).
[16] B. Cockburn and C.WShu "Runge-Kutta Discontinuous Galerkin
Methods for Convection-Dominated Problems" Journal of Scientific
Computing, Vol. 16, No. 3, September 2001 (2001).
[17] B. Cockburn, G.E. Karniadakis and C.-W. Shu. (2000).The development
of discontinuous Galerkin methods. In B. Cockburn, G.E. Karniadakis
and C.-W. Shu (eds), Discontinuous Galerkin Methods. Theory,
Computation and Application, Lecture Notes in Computational Science
and Engineering, Vol. 11, Springer-Verlag, pp. 3-50.
[18] Fareed Ahmed, Faheem Ahmed, Yongheng Guo, andYong Yang, "A
High Order Accurate Nodal Discontinuous Galerkin Method (DGM) for
Numerical Solution of Hyperbolic Equation," International Journal of
Applied Physics and Mathematics vol. 2, no. 5, pp. 362-364, 2012.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:63924", author = "Faheem Ahmed and Fareed Ahmed and Yongheng Guo and Yong Yang", title = "High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation", abstract = "This paper deals with a high-order accurate Runge
Kutta Discontinuous Galerkin (RKDG) method for the numerical
solution of the wave equation, which is one of the simple case of a
linear hyperbolic partial differential equation. Nodal DG method is
used for a finite element space discretization in 'x' by discontinuous
approximations. This method combines mainly two key ideas which
are based on the finite volume and finite element methods. The
physics of wave propagation being accounted for by means of
Riemann problems and accuracy is obtained by means of high-order
polynomial approximations within the elements. High order accurate
Low Storage Explicit Runge Kutta (LSERK) method is used for
temporal discretization in 't' that allows the method to be nonlinearly
stable regardless of its accuracy. The resulting RKDG
methods are stable and high-order accurate. The L1 ,L2 and L∞ error
norm analysis shows that the scheme is highly accurate and effective.
Hence, the method is well suited to achieve high order accurate
solution for the scalar wave equation and other hyperbolic equations.", keywords = "Nodal Discontinuous Galerkin Method, RKDG,
Scalar Wave Equation, LSERK", volume = "6", number = "8", pages = "1223-6", }
