Semi-implicit donor cell differencing and the hyperbolic conservation laws having source terms Original Research Article

Two semi-implicit versions of the first-order Courant, Isaacson and Rees upwind difference scheme for hyperbolic conservation laws having source terms are derived. Fourier analysis is used to analyse stability and phase accuracy of the schemes for a scaler conservation law having a linear source term. Numerical tests, in one and two dimensions, are performed for further insight and justification of their use in case the source term is non-linear. It is shown that the first variant enhances only stability while the second one improves both stability and phase accuracy. The second variant proves to be superior to the original scheme as well as our DOCTR and DOCSI (DOnor Cell differencing using TRapezoidal/SImpsonʹs rule for source term integration) schemes (Ali and Kassar, Int. J. Modern Phys. C, Vol. 5, No. 2, 1994).