On Multiscale Variational and Streamline Diffusion Schemes for a Coupled Nonlinear Telegraph System

References

• Asadzadeh, M., Rostamy, D., Zabihi, F. (2011). A posteriori error estimates for a Coupled Wave System with a Local damping. J. Math. Sci. 175:228–248.
   Google Scholar
• Asadzadeh, M. (1990). Streamline diffusion methods for the Vlasov-Poisson equations. RAIRO Math. Mod. and Numer. Anal. 24:177–196.
   Google Scholar
• Asadzadeh, M. (1997). Streamline diffusion methods for the fermi and Fokker-Planck equations. Transp. Theory Stat. Phys. 26(3):319–340.
   Google Scholar
• Brooks, A., Hughes, T. (1982). Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. FENOMECH’81, Part I (Stuttgart, 1981). Comput. Methods Appl. Mech. Engrg. 32(1–3):199–259.
   Google Scholar
• Ciarlet, P. G. (1987). The Finite Element Method for Elliptic Problems. Amsterdam: North Holland.
   Google Scholar
• Fushchych, W. I., Symenoh, Z. I. (1997). Symmetry of equations with convection terms. Nonlinear Mathematical Physics 4(3–4):470–479.
   Google Scholar
• Johnson, C. (1993). Discontinous Galerkin finite element methods for second order hyperbolic problems. Comput. Methods Appl. Mech. Eng. 107:117–129.
   Google Scholar
• Johnson, C., Saranen, J. (1986). Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Math. Comp. 47(175):1–18.
   Google Scholar
• Jiwari, R., Pandit, S., Mittal, R. C. (2012a). A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. Appl. Math. Comput. 218(13):7279–7294.
   Google Scholar
• Jiwari, R., Pandit, S., Mittal, R. C. (2012b). A differential quadrature algorithm for the numerical solution of the second-order one dimensional hyperbolic telegraph equation. Int. J. Nonlinear Sci. 13(3):259–266.
   Google Scholar
• Leilei Wei, Huiya Dai, Dingling Zhang, Zhiyong Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation Calcolo. DOI: 10. 1007/s10092-013-0084-6.
   Google Scholar
• Li, Y. (2007). Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations. J. Math. Anal. Appl. 327:997–1009.
   Google Scholar
• Lakestani, M., Nemati Saray, B. (2010). Numerical solution of telegraph equation using interpolating scaling functions. Comput. Math. Appl. 60:1964–1972.
   Google Scholar
• Larson, M. G., Målqvist, A. (2007). Adaptive variational multiscale methods based on a posteriori error estimation, Energy norm estimates for elliptic problems. Comput. Methods in Appl. Mech. Eng. 196(21-24):2313–2324.
   Google Scholar
• Ma, R. (2000). Multiple nonnegative solutions of second-order systems of boundary value problems. Nonlinear Anal. 42:1003–1010.
   Google Scholar
• Mawhin, J., Ortega, R., Robles-Perez, A. M. (2000). A maximum principle for bounded solutions of the telegraph equations and applications to nonlinear forcings. J. Math. Anal. Appl. 251:695–709.
   Google Scholar
• Mawhin, J., Ortega, R., Robles-Perez, A. M. (2005). Maximum principles for bounded solutions of the telegraph equation in space dimensions two or three and applications. J. Diff. Eq. 208:42–63.
   Google Scholar
• Pandit, S., Kumar, M., Tiwari, S. (2015). Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients. Comput. Phys. Commun. 187:83–90.
   Google Scholar
• Szepessy, A. (1991). Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions. RAIRO Modl. Math. Anal. Numr. 25(6):749–782.
   Google Scholar
• Wang, F., An, Y. (2009). Existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 349:30–42.
   Google Scholar
• Wang, F., An, Y. (2008). Nonnegative doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 338:91–100.
   Google Scholar
• Wang, F. (2011). Doubly periodic solutions of a coupled nonlinear telegraph system with weak singularies. Nonlinear Anal. Real world appl. 12:254–261.
   Google Scholar