The dissipative Generalized Hydrodynamic equations and their numerical solution

“Generalized Hydrodynamics” (GHD) stands for a model that describes one-dimensional integrable systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic type, where the main unknown, statistical distribution function f(t,z,θ), lives phase space which is constituted by position variable z, and “kinetic” θ, actually wave-vector, called “rapidity”. Two key features are first non-local coupling advection term, second an infinite set conserved quantities, prevent system from thermalizing. To go beyond this, we consider dissipative equations, obtained supplementing right-hand side with diffusion operator Boltzmann-type collision integral. In this paper, deal new high-order numerical methods efficiently solve these equations. particular, devise novel backward semi-Lagrangian solving advective part (the so-called Vlasov equation) using time-Taylor series expansion fields, whose successive time derivatives recursive procedure. This temporal approximation fields used design implicit/explicit Runge–Kutta methods, compared Adams–Moulton schemes. For source terms, operators, use compare different literature.