Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations

2019 ◽  
Vol 35 (6) ◽  
pp. 1971-1999
Author(s):  
Xiuling Hu ◽  
Shanshan Wang ◽  
Luming Zhang
Keyword(s):  
Difference Scheme ◽  
Error Estimates ◽  
Boussinesq Equations ◽  
Maximum Error ◽  
Compact Difference Scheme ◽  
Nonlinear Schrödinger ◽  
Compact Difference

scholarly journals Conservative compact difference scheme based on the scalar auxiliary variable method for the generalized Kawahara equation

2021 ◽  
Author(s):  
Hongtao Chen ◽  
Yuyu He
Keyword(s):  
Difference Scheme ◽  
Error Estimates ◽  
Energy Method ◽  
Auxiliary Variable ◽  
Variable Method ◽  
Compact Difference Scheme ◽  
Kawahara Equation ◽  
Discrete Energy ◽  
Discrete Energy Method ◽  
Compact Difference

In this paper, a conservative compact difference scheme for the generalized Kawahara equation is constructed based on the scalar auxiliary variable (SAV) approach. The discrete conservative laws of mass and Hamiltonian energy and boundedness estimates are studied in detail. The error estimates in discrete $L^{\infty}$-norm and $L^2$-norm of the presented scheme are analyzed by using the discrete energy method. We give an efficiently algorithm of the presented scheme which only needs to solve two decoupled equations.

scholarly journals Convergence of an Eighth-Order Compact Difference Scheme for the Nonlinear Schrödinger Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Tingchun Wang
Keyword(s):  
Schrödinger Equation ◽  
Difference Scheme ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Compact Difference Scheme ◽  
Eighth Order ◽  
Time Step ◽  
Nonlinear Schrödinger ◽  
Compact Difference

A new compact difference scheme is proposed for solving the nonlinear Schrödinger equation. The scheme is proved to conserve the total mass and the total energy and the optimal convergent rate, without any restriction on the grid ratio, at the order of O(h8+τ2) in the discrete L∞-norm with time step τ and mesh size h. In numerical analysis, beside the standard techniques of the energy method, a new technique named “regression of compactness” and some lemmas are proposed to prove the high-order convergence. For computing the nonlinear algebraical systems generated by the nonlinear compact scheme, an efficient iterative algorithm is constructed. Numerical examples are given to support the theoretical analysis.

Direct calculation of waves in fluid flows using high-order compact difference scheme

AIAA Journal ◽  
1994 ◽  
Vol 32 (9) ◽  
pp. 1766-1773 ◽  
Author(s):  
Sheng-Tao Yu ◽  
Lennart S. Hultgren ◽  
Nan-Suey Liu
Keyword(s):  
Difference Scheme ◽  
Fluid Flows ◽  
Direct Calculation ◽  
High Order ◽  
Compact Difference Scheme ◽  
Compact Difference
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