High order of accuracy stable difference schemes for numerical solutions of NBVP for hyperbolic equations

2012 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ozgur Yildirim
Keyword(s):  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
High Order ◽  
Difference Schemes ◽  
Order Of Accuracy ◽  
High Order Of Accuracy

scholarly journals Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations

2005 ◽  
Vol 2005 (2) ◽  
pp. 183-213 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskii
Keyword(s):  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
High Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
Accuracy Difference ◽  
High Order Of Accuracy ◽  
The Stability

We consider the abstract Cauchy problem for differential equation of the hyperbolic typev″(t)+Av(t)=f(t)(0≤t≤T),v(0)=v0,v′(0)=v′0in an arbitrary Hilbert spaceHwith the selfadjoint positive definite operatorA. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of this problem are presented. The stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.

scholarly journals Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskiĭ
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Continuous Functions ◽  
High Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Smooth Functions ◽  
Order Of Accuracy ◽  
High Order Of Accuracy ◽  
The Difference

It is well known the differential equation−u″(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the positive operatorAis ill-posed in the Banach spaceC(E)=C((−∞,∞),E)of the bounded continuous functionsϕ(t)defined on the whole real line with norm‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions inC(τ,E)of these difference schemes is established.

scholarly journals On stable high order difference schemes for hyperbolic problems with the Neumann boundary conditions

2019 ◽  
Vol 9 (1) ◽  
pp. 60-72 ◽  
Author(s):  
Ozgur Yildirim
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Nonlocal Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Order Difference

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.

scholarly journals Simple and High-Accurate Schemes for Hyperbolic Conservation Laws

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Renzhong Feng ◽  
Zheng Wang
Keyword(s):  
Hyperbolic Conservation Laws ◽  
Hyperbolic Equations ◽  
Order Approximation ◽  
High Order ◽  
Special Choice ◽  
Approximation Accuracy ◽  
Order Of Accuracy ◽  
Hyperbolic Conservation ◽  
Spurious Oscillations ◽  
High Order Of Accuracy

The paper constructs a class of simple high-accurate schemes (SHA schemes) with third order approximation accuracy in both space and time to solve linear hyperbolic equations, using linear data reconstruction and Lax-Wendroff scheme. The schemes can be made even fourth order accurate with special choice of parameter. In order to avoid spurious oscillations in the vicinity of strong gradients, we make the SHA schemes total variation diminishing ones (TVD schemes for short) by setting flux limiter in their numerical fluxes and then extend these schemes to solve nonlinear Burgers’ equation and Euler equations. The numerical examples show that these schemes give high order of accuracy and high resolution results. The advantages of these schemes are their simplicity and high order of accuracy.

Sign in / Sign up

Export Citation Format

Share Document