Abstract
In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L 2 stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.
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REFERENCES
Bassi, F., and 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, 267–279.
Biswas, R., Devine, K. D., and Flaherty, J. (1994). Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. 14,255–283.
Cockburn, B. (1999). Discontinuous Galerkin methods for convection-dominated problems. In Barth, T. J., and Deconinck, H. (eds.), High-Order Methods for Computational Physics, Lecture Notes in Computational Science and Engineering, Vol. 9, Springer, pp. 69–224.
Cockburn, B., Hou, S., and Shu, C.-W. (1990). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comp. 54, 545–581.
Cockburn, B., Karniadakis, G., and Shu, C.-W. (2000). The development of discontinuous Galerkin methods. In Cockburn, B., Karniadakis, G., and Shu C.-W. (eds.), Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer, Part I: Overview, pp. 3–50.
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. J. Comput. Phys. 84, 90–113.
Cockburn, B., Luskin, M., Shu, C.-W., and Süli, E. Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comp. To appear.
Cockburn, B., and Shu, C.-W. (1991). The Runge-Kutta local projection P1-discontinuous-Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. (M2AN) 25, 337–361.
Cockburn, B., and Shu, C.-W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework. Math. Comp. 52, 411–435.
Cockburn, B., and Shu, C.-W. (1998). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws V: Multidimensional systems. J. Comput. Phys. 141, 199–224.
Cockburn, B., and Shu, C.-W. (1998). The local discontinuous Galerkin method for time-dependent convection diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463.
Cockburn, B., and Shu, C.-W. (2001). Runge-Kutta Discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261.
Dekker, K., and Verwer, J. G. (1984). Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland.
Jiang, G.-S., and Shu, C.-W. (1994). On cell entropy inequality for discontinuous Galerkin methods. Math. Comp. 62, 531–538.
Johnson, C., and Pitkäranta, J. (1986). An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46, 1–26.
LeVeque, R. J. (1990). Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel.
Shu, C.-W. (1987). TVB uniformly high-order schemes for conservation laws. Math. Comp. 49, 105–121.
Shu, C.-W. (2001). Different formulations of the discontinuous Galerkin method for the viscous terms. In Shi, Z.-C., Mu, M., Xue W., and Zou, J. (eds.), Advances in Scientific Computing, Science Press, pp. 144–155.
Shu, C.-W., and Osher, S. (1988). Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471.
Yan, J., and Shu, C.-W., A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. to appear.
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Yan, J., Shu, CW. Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives. Journal of Scientific Computing 17, 27–47 (2002). https://doi.org/10.1023/A:1015132126817
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DOI: https://doi.org/10.1023/A:1015132126817