In this paper, a localized space-time method of fundamental solutions (LSTMFS) is proposed to solve the diffusion and convection-diffusion problems. The proposed LSTMFS only requires some arbitrarily-distributed nodes inside the space-time domain and along its boundary. The local subdomain corresponding to each node can firstly be determined based on the Euclidean distance between the nodes. Then, the variable at each node can be expressed as a linear combination of variables at its supporting nodes. By solving a resultant sparse system, the variable at any node in the considered space-time domain can be obtained. Compared with the traditional space-time method of fundamental solutions, the proposed LSTMFS is more suitable for solving large-scale and long-time diffusion problems. Furthermore, the LSTMFS without temporal-difference is simple, accurate and easy-to-implement due to its semi-analytical and meshless features. Numerical experiments, including diffusion and convection-diffusion problems, confirm the validity and accuracy of the proposed LSTMFS.
Localized space-time method of fundamental solutions, meshless method, time-dependent fundamental solutions, diffusion, convection-diffusion.