Algorithm

Mathematical optimization

Numerical integration

Mathematical analysis

Backward differentiation formula

Economics

Political science

Runge–Kutta methods

Norm (philosophy)

Collocation method

Bandwidth (computing)

Convergence (economics)

Applied mathematics

Maximum principle

Computer network

Mathematics

Computer science

Computation

Numerical analysis

Economic growth

Ordinary differential equation

Variational integrator

Discretization

Differential equation

Integrator

Optimal control

Applied Numerical Mathematics

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn equation has been an open problem in recent years. This work provides a positive answer by designing and analyzing a class of up to fourth order maximum principle preserving integrators for the Allen-Cahn equation. First, the second order finite difference discretization is applied to the Allen-Cahn equation in the space direction. The obtained semi-discrete system also preserves the maximum principle and the energy dissipation law. Then the fully discrete numerical scheme is obtained by applying the Lawson transformation and the Runge-Kutta integration in the time direction. We define sufficient conditions for explicit integration factor Runge-Kutta scheme to preserve the maximum principle, namely, the Shu-Osher form of the underlying Runge-Kutta scheme has non-negative coefficients αi,j, nondecreasing abscissas ci and the time step size τ>0 satisfies τ{βi,jαi,j}∈[−4,12]. We prove that the proposed method is convergent with order O(τp+h2) in the discrete L∞ norm. A fast solver is then applied to the discrete system to accelerate numerical computations. Various experiments for 1D, 2D and 3D problems are provided to illustrate the high-order convergence and maximum principle preserving of the proposed algorithms over a long time and verify the theoretical analysis.

Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation