Unconditional and optimal H 2 -error estimates of two linear and conservative finite difference schemes for the Klein-Go
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Unconditional and optimal H2 -error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schr¨odinger equation in high dimensions Tingchun Wang1 · Xiaofei Zhao2 · Jiaping Jiang1
Received: 30 March 2014 / Accepted: 28 July 2017 © Springer Science+Business Media, LLC 2017
Abstract The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear KleinGordon-Schr¨odinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H 2 -error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h2 + τ 2 ) with mesh-size h and time step τ in the discrete H 2 -norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.
Communicated by: Ivan Oseledets Tingchun Wang
[email protected] Xiaofei Zhao [email protected] Jiaping Jiang [email protected] 1
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2
INRIA-Rennes Bretagne Atlantique and IRMAR, Universit´e de Rennes 1, Rennes, France
T. Wang et al.
Keywords Klein-Gordon-Schr¨odinger equation · Finite difference method · Solvability · Energy conservation · H 2 convergence · Optimal error estimates Mathematics Subject Classification (2010) Primary 65M06 · 65M12
1 Introduction This paper aims to analyze two finite difference time domain schemes for solving the dimensionless Klein-Gordon-Schr¨odinger (KGS) equations 1 i∂t ψ(x, t) + ψ(x, t) + φ(x, t)ψ(x, t) = 0, x ⊂ Rd , t > 0, 2 ∂tt φ(x, t) − φ(x, t) + μ2 φ(x, t) = |ψ(x, t)|2 , x ⊂ Rd , t > 0, (ψ, φ, ∂t φ)(x, 0) = (ψ0 , φ0 , φ1 )(x), x ∈ Rd ,
(1.1) (1.2) (1.3)
which is the classical model representing the dynamics of conserved complex nucleon fields ψ interacting with neutral real scalar meson fields φ. See [6, 9, 20] for its derivation and non-dimensionalization. Here, t is time, x = (x, y) in two dimensions (2D), i.e. d = 2, and respectively x = (x, y, z) in three dimensions (3D), i.e. d = 3, are the Cartesian coordinates, μ describes the ratio of mass between a meson and a nucleon, ψ0 is a given complex-valued function and φ0 and φ1 are two given real-valued functions. It is clear to see that the KGS equations (1.1)–(1.3) conserve the total mass, M(t) := Rd |ψ(x, t)|2 dx ≡ M(0) := Rd |ψ0 (x)|2 dx, t > 0, (1.4) and the total energy 1 2
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