The authors consider a linear three-dimensional, time-dependent convection-diffusion equation, together with its transformed version where the convection term has been eliminated (thus, a conduction equation). The discretized equations related to these partial differential equations provide basically a linear relation between the value at a central nodal point and those of its neighbours in both space and time. Here the weights, i.e. the coefficients in that relation, are determined to ensure that dispersive terms (in the Taylor expansion of the discretization error) are cancelled out. In this way second-order explicit and implicit schemes, fourth-order strongly and weakly implicit schemes, temporally first-order schemes, and Crank-Nicholson implicit schemes are established. Extensive numerical tests (and comparisons against analytical benchmarks as that of J. D. Cole [Quart. Appl. Math. 9, 225-236 (1951; Zbl 0043.09902)], or of E. Hopf [Commun. Pure Appl. Math. 3, 201-230 (1950; Zbl 0039.10403)], or of C. A. J. Fletcher [Int. J. Numer. Methods Fluids, 3, 213-216 (1983; Zbl 0563.76082)] are done and sketched by nowadays customary graphics packages. Finally, the numerical influence of physical and artificial parameters is discussed – a typical example of experimental mathematics.