Fast Low-Rank Solution of the Multidimensional Hyperbolic Problems

Аннотация

In this paper, the numerical solution of multidimensional hyperbolic problems is discussed with the quantized tensor train (QTT)-approximation methods. Three schemes are proposed. First, an improved implicit time iteration scheme is presented by using the two-site density matrix renormalization group (DMRG) algorithm to solve a linear system at each time step. Second, the time is considered as an independent dimension, and a discretization of the whole differential equation is introduced with all spatial and time dimensions connected in one big global linear system. Then the problem is solved in the QTT-format. The third scheme is to solve the global system by splitting the global time interval into several subintervals. The numerical experiments, with these three schemes applied to the wave equation, show that the complexity of the first scheme is linear while that of the second and third schemes is log-linear in both time and spatial grid points.