We theoretically and computationally investigate long-memory processes based on the Markovian lifts of affine jump-diffusion processes. A nominal superposition process consisting of an infinite number of interacting affine processes is considered, along with its finite-dimensional version and associated generalized Riccati equations. We propose a splitting scheme suited to the Markovian lifts where jump and diffusion parts are dealt with separately based on recently developed exact discretization methods. We examine the computational performance of the scheme through comparisons with the analytical results. We also numerically investigate a more complex model arising in the environmental sciences and some extended cases in which superposed processes belong to a class of nonlinear processes that generalize affine processes.