english An International Journal of Optimization and Control: Theories & Applications ISSN:2146-0957 eISSN:2146-5703 Vol.9, No.1, pp.60-72 (2019) http://doi.org/10.11121/ijocta.01.2019.00592 RESEARCH ARTICLE On stable high order difference schemes for hyperbolic problems with the Neumann boundary conditions Ozgur Yildirim Department of Mathematics, Yildiz Technical University, Turkey ozgury@yildiz.edu.tr ARTICLE INFO ABSTRACT Article History: Received 16 March 2018 Accepted 01 November 2018 Available 31 January 2019 In this paper, third and fourth order of accuracy stable difference schemes for approximately solving multipoint nonlocal boundary value problems for hyperbolic equations with the Neumann boundary conditions are considered. Stability estimates for the solutions of these difference schemes are presented. Finite difference method is used to obtain numerical solutions. Numerical results of errors and CPU times are presented and are analyzed. Keywords: Nonlocal and multipoint BVPs Stability Abstract hyperbolic equations Finite difference methods AMS Classification 2010: 34B10, 34D20, 35L90, 65N06 1. Introduction Many mathematical models of natural and ap- plied sciences phenomena such as fluid mechan- ics, hydrodynamics, electromagnetics and various areas of physics are based on hyperbolic partial differential equations. Modeling some of these phenomena, imposing nonlocal conditions may be more accurate than classical conditions. Nonlo- cal boundary condition is a relation between the values of unknown function on the boundary and inside of the given domain. Over the last decades, boundary value problems with nonlocal boundary conditions have become a rapidly growing area of research. Such types of boundary conditions are encountered in applications including thermoelas- ticity [1], climate control systems [2] and financial mathematics [3]. Boundary value problems for parabolic, elliptic and equations of mixed types are actively studied by many scientists for decades (see [4]- [27]). Stability has been an important re- search area in the development of numerical meth- ods. Particulary, in this work stability analysis is performed by suitable unconditionally stable difference schemes with an unbounded operator. Some results of this paper, without proof, are pre- sented in [27]. In the present paper, third and fourth order of accuracy stable difference schemes for approxi- mately solving the multipoint nonlocal boundary value problem (NBVP)                    ∂ 2 u(t,x) ∂t 2 - m ∑ r=1 (a r (x)u xr ) xr = f (t,x), x =(x 1 ,...,x m ) ∈ Ω, 0 <t< 1, u(0,x)= n ∑ j =1 α j u (λ j ,x)+ ϕ(x),x ∈ Ω, u t (0,x)= n ∑ j =1 β j u t (λ j ,x)+ ψ(x),x ∈ Ω (1) for the multidimensional hyperbolic equation with the Neumann boundary condition ∂u(t,x) ∂n | x∈S =0,x ∈ S or mixed conditions u(t,x)| x∈S 1 =0, ∂u(t,x) ∂n | x∈S 2 =0, 60