Generalized Square Conservative Multistep Finite Difference Scheme Incorporating Historical Observations

In this paper, a multistep finite difference scheme has been proposed, whose coefficients are determined taking into consideration compatibility and generalized quadratic conservation, as well as incorporating historical observation data. The schemes have three advantages: high-order accuracy in time, generalized square conservation, and smart use of historical observations. Numerical tests based on the one-dimensional linear advection equations suggest that reasonable consideration of accuracy, square conservation, and inclusion of historical observations is critical for good performance of a finite difference scheme.