A HIGHLY EFFICIENT AND ACCURATE NEW SCALAR AUXILIARY VARIABLE APPROACH FOR GRADIENT FLOWS

We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit linear terms, we are able to reduce the number of linear equations with constant coefficients to be solved at each time step from two to one, so the computational cost of the new SAV approach is essentially half of the original SAV approach while keeping all its other advantages. This technique is also extended to the multiple SAV approach. Secondly, instead of discretizing the dynamical equation for the auxiliary variable, we use a first-order approximation of the energy balance equation, which allows us to construct high-order unconditionally energy-stable SAV schemes with uniform and, more importantly, variable time step sizes, enabling us to construct, for the first time, high-order unconditionally stable adaptive time-stepping backward differentiation formula schemes. Representative numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.