Three-Layer Compact Difference Scheme for a Hyperbolic Heat Conduction Equation

On the basis of hyperbolic heat conduction equation, three-layer compact difference schemes of orders 4 + 2, 4 + 4 and a Saul’ev scheme of order 6 + 3 are constructed on minimal three-point stencils in space. The close relationship between the explicit Chetverushkin scheme and the three-layer asymptotically stable Samarskii scheme is shown. It is also proposed to combine the classical models of filtration and heat conduction into a single mathematical model based on the definition of a generalized solution according to Godunov. The algorithms of order 4 + 2 obtained in this way are generalized to quasi–linear parabolic equations with arbitrary nonlinearity. Numerical calculations of a number of test problems are given, illustrating the efficiency of compact schemes.