Semi-implicit methods for advection equations with explicit forms of numerical solution

We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation for which the numerical solutions can be obtained in a fixed number of forward and backward alternating substitutions. The methods use a novel combination of implicit and explicit time discretizations for one-dimensional case and the Strang splitting method in several dimensional case. The methods are described for advection equations with a continuous variable velocity that can change its sign inside of computational domain. The methods are unconditionally stable in the non-conservative case for variable velocity and for variable numerical parameter. Several numerical experiments confirm the advantages of presented methods including an involvement of differential programming to find optimized values of the variable numerical parameter.