A high order difference scheme is established for two-dimensional linear hyperbolic equation with Neumann boundary conditions. The third and fifth derivatives of solution at the boundary can be got by using the boundary conditions and the equation, then the nine points, six points and four points compact difference schemes are respectively established at the inner points of the region, inner points and corner points of the boundary by using the finite difference method. To obtain the convergence and stability of the numerical solution in maximum norm, a new norm is introduced to estimate maximum norm. Then two priori estimates of the difference scheme are shown and convergence and stability are derived. The convergence order of the difference scheme in maximum norm is O(τ2 + h4) where tau and h are temporal and spatial step size, respectively. Some numerical examples illustrate the convergence of the high order difference schemes presented in this paper.
. A HIGH ORDER DIFFERENCE SCHEME FOR TWO-DIMENSIONAL LINEAR HYPERBOLIC EQUATION WITH NEUMANN BOUNDARY CONDITIONS[J]. Mathematica Numerica Sinica, 2019, 41(3): 266-294.
Dehghan M, Shokri A. A meshless method for numerical solution of a linear hyperbolic equation with variable coffcients in two space dimensional[J]. Numer. Methods Partial Differential Equations, 2009, 25:494-506.
He Dongdong. An unconditionally stable spatial sixth-order CCD-ADI method for the two dimensionallinear telegraph equation[J]. Numer. Algorithms, 2016, 72(4):1103-1117.
Mohanty R K, Jain M K. An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation[J]. Numer. Methods Partial Differential Equations, 2001, 17(6):684-688.
Liu J, Tang K. A new unconditionally stable ADI compact scheme for the two-space dimensional linear hyperbolic equation[J]. Internat. J. Comput. Math., 2010, 87(10):2259-2267.
Evans D J, Bulut H. The numerical solution of Burgers' equation by the alternating group explicit (age) method[J]. Internat. J. Comput. Math., 2003, 29(1):1289-1297.
Hu Y Y, Liu H W. An unconditionally stable spline difference scheme for solving the second 2D linear hyperbolic equation[C]. Computer Modelling and Simulation International Conference, on (2010), Sanya China, 375-378.
Mehdi Dehghan, Akbar Mohebbi. The combination of collocation, fnite difference, and multigrid methods for solution of the two-dimensional wave equation[J]. Numer. Methods Partial Differential Equations, 2008, 24(3):897-910.
Mohanty R K. An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions[J]. Appl. Math. Comput., 2004, 152(3):799-806.
Dehghan M, Ghesmati A. Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method[J]. Eng. Anal. Bound. Elem., 2010, 34(1):51-59.
Pekmen B, Tezer-Sezgin M. Differential quadrature solution of hyperbolic telegraph equation[J]. J. Appl. Math., 2012, 2012, Article ID 924765, 18 pages.
Ram Jiwari, Sapna Pandit, Mittal R C. A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions[J]. Appl. Math. Comput., 2012, 218(13):7279-7294.
Dehghan M, Ghesmati A. Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation[J]. Eng. Anal. Bound. Elem., 2010, 34(4):324-336.
Heinz-Otto Kreiss, Anders Petersson N, Jacob Ystrom. Difference approximations of the Neumann problem for the second order wave equation[J]. SIAM J. Numer. Anal., 2004, 42(3):1292-1323.
Appelo D, Petersson N A. A fourth-order accurate embedded boundary method for the wave equation[J]. SIAM J. Sci. Comput., 2012, 34(6):A2982-A3008.
Gao Guang-Hua, Sun Zhi-Zhong. Compact difference schemes for heat equation with Neumann boundary conditions (Ⅱ)[J]. Numer. Methods Partial Differential Equations, 2013, (29):1459-1486.
Sun Z Z. Compact difference schemes for heat equation with Neumann boundary conditions[J]. Numer. Methods Partial Differential Equations, 2010, 25(6):1320-1341.
Zhou Y L. Application of discrete functional analysis to the fnite difference methods[M]. Interna-tional Academic Publishers, 1990, 8(1):49-65.