On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations 论文

摘要

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 4"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -regular solution, a first-order error bound in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm is shown and used to derive a second-order error bound in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">L_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm. For the cubic Schrödinger equation with an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 4"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -regular solution, first-order convergence in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm is used to obtain second-order convergence in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">L_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript m"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">H^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conditional stability for error propagation, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the Schrödinger-Poisson system and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 2"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the cubic Schrödinger equation.