We present an innovative method based on the linear advection equation, an important partial differential equation, to perform the fusion of images. The basic idea of this method is to insert the relevant information from other source images into the current source image through an advection process. Furthermore, we present the discrete scheme of this model and compare it with classical fusion approaches, the diffusion equation-based method, and some state-of-the-art fusion approaches on three groups of fusion images that are often used in the image fusion research. The results of experiments show that the fusion method based on the advection equation is comparable with the best of the classical, diffusion-based, and state-of-the-art methods. The high “weighted performance metric” of fused images certifies that the relevant information is well injected from the input to the output images. Moreover, this method has fewer adjustable parameters with settings that affect the metric less than other methods, and the evolution from input to output is also faster than the diffusion-based method. In addition, this model allows us to cope with noisy source image fusion by adding a diffusion term in the equation, thereby combining the denoising process with the fusion process.