Compact ADI Difference Scheme for the 2D Time Fractional Nonlinear Schrödinger Equation

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn><mo>−</mo><mn>2</mn><mo>−</mo><mn>3</mn></mrow></semantics></math></inline-formula> formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>τ</mi><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow></msup><mo>+</mo><msubsup><mi>h</mi><mrow><mi>x</mi></mrow><mn>4</mn></msubsup><mo>+</mo><msubsup><mi>h</mi><mrow><mi>y</mi></mrow><mn>4</mn></msubsup><mo>)</mo></mrow><mrow><mo>(</mo><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is obtained by adding a small term, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>h</mi><mi>x</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>h</mi><mi>y</mi></msub></semantics></math></inline-formula> are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme.