This project can be divided into two parts. The goal of the first part is to numerically implement the Cahn-Hilliard equation in one dimension both explicitly and implicitly. This will be done using Matlab. The goal of the second part is to validate the coupled Cahn-Hilliard-Navier-Stokes equation and the dynamic boundary condition for moving contact lines of (Carlson et al, 2011, p.9) by considering a two-dimensional spreading droplet case. This will be done using the CFD software OpenFOAM. In Chapter 1, the theory of positive and negative diffusion, including the normal diffusion equation and the Cahn-Hilliard equation, are discussed. Some background is given regarding the thermodynamics of the Cahn-Hilliard equation and its steady-state solution. After that, the theory of the coupled Cahn-Hilliard-Navier-Stokes equation, the dynamic boundary condition for moving contact lines and the case which is implemented in OpenFOAM, are discussed. In Chapters 2 and 3, the diffusion equation and the Cahn-Hilliard equation are implemented in one dimension, using the Euler Forward scheme. In implementing the Cahn-Hilliard equation, two different discretizations are used, of which only one gives the desired results. Next, an extensive stability analysis is done, using a linearization of the Cahn-Hilliard equation as well as numerical experiments. The stability condition is increasingly severe with increasing interface width. Regarding the results of the evolving interface, a qualitative analysis is done which discusses three subjects: the deviation of the solutions with the steady-state solution, the interface width for different parameters and grid sizes and the interface overshoot, which is an unphysical appearence. In Chapter 4, two semi-implicit methods and one implicit iterative method, are discussed. The implementations of the two semi-implicit methods, Implicit-Explicit (ImEx) and Modified Furihata, are succesful and their stability conditions are better than the stability condition of the Euler Forward scheme, for most interface widths. The results regarding the evolving interface are nearly identical to the results of the Euler Forward scheme, therefore the qualitative analysis is also similar. The implicit iterative method, which involves the use of the G\^ateaux derivative, has not been succesfully implemented, eventhough two different discretizations are used. The results regarding the evolving interface are behaving in a positive diffusive way, which results in a flattening interface with time. In Chapter 5, the coupled Cahn-Hilliard-Navier-Stokes equation and a dynamic boundary condition for moving contact lines are used to model a spreading droplet on a flat surface. The implemented model is validated using different cases in which the steady-state contact angle and the friction factor of the surface varies. Next, parametric studies are done regarding the interface width, the surface tension and the ratio of the surface tension and the friction factor. The conclusions are that the modeled system differs too much from the system in literature to make an absolute comparison but, qualitatively, the model behaves as expected.