An operator‐splitting algorithm for two‐dimensional convection–dispersion–reaction problems

An operator‐splitting algorithm for the two‐dimensional convection–dispersion–reaction equation is developed. The flow domain is discretized into triangular elements which are fixed in time. The governing equation is split into three successive initial value problems: a pure convection problem, a pure dispersion problem and a pure reaction problem. For the pure convection problem, solutions are found by the method of characteristics. The solution algorithm involves tracing the characteristic lines backwards in time from a vertex of an element to an interior point. A cubic polynomial is used to interpolate the concentration and its derivatives on an element. For the pure dispersion problem, an explicit finite element algorithm is employed. Analytical solutions are obtained for the pure reaction problem. The treatment of the boundary conditions is also discussed. Several numerical examples are presented. Numerical results agree well with analytical solutions. Because cubic polynomials are used in the interpolation, very little numerical damping and oscillation are introduced, even for the pure convection problem.