An efficient variable step numerical method for the three-dimensional nonlinear evolution equation

This work presents a second-order in time variable step BDF2 ADI orthogonal spline collocation (OSC) scheme for the three-dimensional nonlinear fractional evolution equation. Non-uniform meshes can overcome the difficulties arising from the singularity of the exact solution at \(t = 0\) $t=0$ . The time-stepped second-order backward differentiation formula (BDF2) is utilised in the temporal direction, and the spatial direction combines the alternating direction implicit (ADI) method and the arbitrary order OSC method. Then we prove the stability of the numerical scheme and give error estimates. The theoretical analysis shows that the proposed numerical scheme based on variable time step can solve the singularity problem effectively. Finally, numerical examples are given to verify our theoretical analysis.