seminar

Langevin method is an important method to calculate path integrals in quantum chromodynamics, which requires to solve the Langevin equations on the special unitary group SU(n) or the special linear group SL(n, C). Since the  Langevin equations are nonlinear stochastic differential equations, we focus on the research of numerical methods to solve them. With the Ito-Taylor expansions of the equations, the truncated Taylor schemes do not guarantee that the solutions are on SU(n) or SL(n, C).

Based on the exponential expansions, the new high-order numerical schemes are presented through introducing the correction matrices, which overcomes the shortcoming of the truncated Taylor schemes successfully. In order to improve the efficiency, we also construct the Runge-Kutta schemes with the approximations of the derivatives. The numerical experiments show all of the schemes reach the theoretical convergence order.