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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>O</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mn>8</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math> in the discrete <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:mrow></mml:math>-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.

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