The authors present a conservative machine learning-based semi-Lagrangian finite difference (SL FD) method for simulating transport equations. The key aspects of the proposed approach are:
It builds upon the authors' previous work on a conservative machine learning-based SL finite volume (FV) method, but aims to further improve efficiency and simplify implementation.
The method employs an Encode-Process-Decode framework, where a novel dynamical graph neural network (GNN) is used in the processor to handle the complexities associated with tracking upstream points along characteristics.
The encoder maps the solution and normalized shifts to node embeddings, which are then processed by the GNN to learn the optimal SL FD discretization. The decoder ensures exact local mass conservation.
The proposed method achieves unconditional stability and allows for large time step evolution, in contrast to traditional SL FD schemes which are restricted by a CFL condition.
The authors extend the method to solve the nonlinear Vlasov-Poisson system by integrating it with high-order Runge-Kutta exponential integrators, resulting in a novel data-driven conservative SL FD Vlasov-Poisson solver without operator splitting.
Numerical results on benchmark 1D and 2D linear transport equations as well as the nonlinear Vlasov-Poisson system demonstrate the effectiveness and efficiency of the proposed approach compared to traditional numerical schemes.
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by Yongsheng Ch... at arxiv.org 05-06-2024
https://arxiv.org/pdf/2405.01938.pdfDeeper Inquiries