Physics

Energy (signal processing)

Cahn–Hilliard equation

Allen–Cahn equation

Phase field models

Mathematics

Economic growth

Partial differential equation

Statistics

Epistemology

Mathematical analysis

Philosophy

Runge–Kutta methods

Property (philosophy)

Economics

Geometry

Convergence (economics)

Phase (matter)

Numerical analysis

Quantum mechanics

Discretization

Applied mathematics

Scalar (mathematics)

SIAM Journal on Scientific Computing

We construct and analyze a class of extrapolated and linearized Runge--Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the Allen--Cahn and Cahn--Hilliard phase field equations, based on the scalar auxiliary variable (SAV) formulation. We prove that the proposed $q$-stage RK--SAV methods have $q$th-order convergence in time and satisfy a discrete version of the energy decay property. Numerical examples are provided to illustrate the discrete energy decay property and accuracy of the proposed methods.

Energy-Decaying Extrapolated RK--SAV Methods for the Allen--Cahn and Cahn--Hilliard Equations