A second-order TVD implicit–explicit finite volume method for time-dependent convection-reaction equations

A class of conservative methods is developed in the more general framework of cell-centered upwind differences to approximate numerically the solution of one-dimensional non-linear conservation laws with (possibly) stiff reaction source terms. These methods are based on a non-oscillatory piecewise linear polynomial representation of the discrete solution within any mesh interval to compute pointwise solution values. The piecewise linear approximate solution is obtained by approximating the cell average of the analytical solution and the solution slope in every mesh cell. These two quantities are evolved in time by solving a set of discrete equations that are suitably designed to ensure formal second-order consistency. Several numerical tests which are taken from literature illustrate the performance of the method in solving non-stiff and stiff convection-reaction equations in conservative form.