Totally Volume Integral of Fluxes for Discontinuous Galerkin Method (TVI-DG) I-Unsteady Scalar One Dimensional Conservation Laws

Authors

  • Ibrahim. M. Rustum Department of Mechanical Engineering, University of Benghazi, Libya
  • ElHadi. I. Elhadi Department of Mechanical Engineering, University of Benghazi, Libya

DOI:

https://doi.org/10.54172/mjsc.v32i1.124

Keywords:

Scalar conservation laws, Higher order methods, Discontinuous Galerkin, Divergence theorem

Abstract

The volume integral of Riemann flux in the discontinuous Galerkin (DG) method is introduced in this paper. The boundaries integrals of the fluxes (Riemann flux) are transformed into volume integral. The new family of DG method is accomplished by applying divergence theorem to the boundaries integrals of the flux. Therefore, the (DG) method is independent of the boundaries integrals of fluxes (Riemann flux) at the cell (element) boundaries as in classical (DG) methods. The modified streamline upwind Petrov-Galerkin method is used to capture the oscillation of unphysical flow for shocked flow problems. The numerical results of applying totally volume integral discontinuous Galerkin method (TVI-DG) are presented to unsteady scalar hyperbolic equations (linear convection equation, inviscid Burger's equation and Buckley-Leverett equation) for one dimensional case. The numerical finding of this scheme is very accurate as compared with other high order schemes as the weighted compact finite difference method WCOMP.

Downloads

Download data is not yet available.

References

Cockburn, B. (2001). Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. Journal of Computational and Applied Mathematics 128(1):187-204. DOI: https://doi.org/10.1016/S0377-0427(00)00512-4

Cockburn, B., Hou S., and Shu C.-W. (1990). The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Mathematics of Computation 54(190):545-581. DOI: https://doi.org/10.1090/S0025-5718-1990-1010597-0

Cockburn, B., Lin S.-Y., and Shu C.-W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. Journal of Computational Physics 84(1):90-113. DOI: https://doi.org/10.1016/0021-9991(89)90183-6

Cockburn, B., and Shu C.-W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Mathematics of Computation 52(186):411-435. DOI: https://doi.org/10.1090/S0025-5718-1989-0983311-4

Gao, H., and Wang Z. (2009). A high-order lifting collocation penalty formulation for the Navier-Stokes equations on 2-D mixed grids. ratio, 1, 2. DOI: https://doi.org/10.2514/6.2009-3784

Huynh, H. T. (2007). A flux reconstruction approach to high - order schemes including discontinuous Galerkin methods. AIAA paper, 2007- 4079. DOI: https://doi.org/10.2514/6.2007-4079

Liu, L., Li X., and Hu F. Q. (2010). Nonuniform time - step Runge–Kutta discontinuous Galerkin method for computational aeroacoustics. Journal of Computational Physics 229 (19) : 6874-6897. DOI: https://doi.org/10.1016/j.jcp.2010.05.028

Wang, R., Feng H., and Spiteri R. J. (2008). Observations on the fifth - order WENO method with non - uniform meshes. Applied Mathematics and Computation 196(1):433-447. DOI: https://doi.org/10.1016/j.amc.2007.06.024

Wang, Z., and Gao H. (2009). A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. Journal of Computational Physics 228(21):8161-8186. DOI: https://doi.org/10.1016/j.jcp.2009.07.036

Xin, J., and Flaherty J. E. (2006). Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws. Applied Numerical Mathematics 56(3-4):444-458. DOI: https://doi.org/10.1016/j.apnum.2005.08.001

Zhang, S., Jiang S., and Shu C.-W. (2008). Development of nonlinear weighted compact schemes with increasingly higher order accuracy. Journal of Computational Physics 227(15):7294-7321. DOI: https://doi.org/10.1016/j.jcp.2008.04.012

Downloads

Published

2017-06-30

How to Cite

Rustum, I. M., & Elhadi, E. I. (2017). Totally Volume Integral of Fluxes for Discontinuous Galerkin Method (TVI-DG) I-Unsteady Scalar One Dimensional Conservation Laws. Al-Mukhtar Journal of Sciences, 32(1), 36–45. https://doi.org/10.54172/mjsc.v32i1.124

Issue

Section

Research Articles

Categories