Recent works have shown that neural networks are promising parameter-free limiters for a variety of numerical schemes(Morgan et al.in A machine learning approach for detect-ing shocks with high-order hydrodynamic meth...Recent works have shown that neural networks are promising parameter-free limiters for a variety of numerical schemes(Morgan et al.in A machine learning approach for detect-ing shocks with high-order hydrodynamic methods.et al.in J Comput Phys 367:166-191.,2018;Veiga et al.in European Conference on Computational Mechanics andⅦEuropean Conference on Computational Fluid Dynamics,vol.1,pp.2525-2550.ECCM.,2018).Following this trend,we train a neural network to serve as a shock-indicator function using simulation data from a Runge-Kutta discontinuous Galer-kin(RKDG)method and a modal high-order limiter(Krivodonova in J Comput Phys 226:879-896.,2007).With this methodology,we obtain one-and two-dimensional black-box shock-indicators which are then coupled to a standard limiter.Furthermore,we describe a strategy to transfer the shock-indicator to a residual distribution(RD)scheme without the need for a full training cycle and large data-set,by finding a mapping between the solution feature spaces from an RD scheme to an RKDG scheme,both in one-and two-dimensional problems,and on Cartesian and unstruc-tured meshes.We report on the quality of the numerical solutions when using the neural network shock-indicator coupled to a limiter,comparing its performance to traditional lim-iters,for both RKDG and RD schemes.展开更多
Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classica...Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.展开更多
文摘Recent works have shown that neural networks are promising parameter-free limiters for a variety of numerical schemes(Morgan et al.in A machine learning approach for detect-ing shocks with high-order hydrodynamic methods.et al.in J Comput Phys 367:166-191.,2018;Veiga et al.in European Conference on Computational Mechanics andⅦEuropean Conference on Computational Fluid Dynamics,vol.1,pp.2525-2550.ECCM.,2018).Following this trend,we train a neural network to serve as a shock-indicator function using simulation data from a Runge-Kutta discontinuous Galer-kin(RKDG)method and a modal high-order limiter(Krivodonova in J Comput Phys 226:879-896.,2007).With this methodology,we obtain one-and two-dimensional black-box shock-indicators which are then coupled to a standard limiter.Furthermore,we describe a strategy to transfer the shock-indicator to a residual distribution(RD)scheme without the need for a full training cycle and large data-set,by finding a mapping between the solution feature spaces from an RD scheme to an RKDG scheme,both in one-and two-dimensional problems,and on Cartesian and unstruc-tured meshes.We report on the quality of the numerical solutions when using the neural network shock-indicator coupled to a limiter,comparing its performance to traditional lim-iters,for both RKDG and RD schemes.
文摘Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.
