(untitled) (Communications in Mathematical Research, 2021)
In this work, we aim to show how to solve the continuous-time and continuous-space Krause model by using high-order finite difference (FD) schemes. Since the considered model admits solutions with δ-singularities, the FD method cannot be applied directly. To deal with the annoying δ-singularities, we propose to lift the solution space by introducing a splitting method, such that the δ-singularities in one spatial direction become step functions with discontinuities. Thus the traditional shock-capturing FD schemes can be applied directly. In particular, we focus on the two-dimensional case and apply a fifth-order weighted nonlinear compact scheme (WCNS) to illustrate the validity of the proposed method. Some technical details for implementation are also presented. Numerical results show that the proposed method can capture δ-singularities well, and the obtained number of delta peaks agrees with the theoretical prediction in the literature.