68 research outputs found
Invariant preservation in machine learned PDE solvers via error correction
Machine learned partial differential equation (PDE) solvers trade the
reliability of standard numerical methods for potential gains in accuracy
and/or speed. The only way for a solver to guarantee that it outputs the exact
solution is to use a convergent method in the limit that the grid spacing
and timestep approach zero. Machine learned solvers,
which learn to update the solution at large and/or , can
never guarantee perfect accuracy. Some amount of error is inevitable, so the
question becomes: how do we constrain machine learned solvers to give us the
sorts of errors that we are willing to tolerate? In this paper, we design more
reliable machine learned PDE solvers by preserving discrete analogues of the
continuous invariants of the underlying PDE. Examples of such invariants
include conservation of mass, conservation of energy, the second law of
thermodynamics, and/or non-negative density. Our key insight is simple: to
preserve invariants, at each timestep apply an error-correcting algorithm to
the update rule. Though this strategy is different from how standard solvers
preserve invariants, it is necessary to retain the flexibility that allows
machine learned solvers to be accurate at large and/or .
This strategy can be applied to any autoregressive solver for any
time-dependent PDE in arbitrary geometries with arbitrary boundary conditions.
Although this strategy is very general, the specific error-correcting
algorithms need to be tailored to the invariants of the underlying equations as
well as to the solution representation and time-stepping scheme of the solver.
The error-correcting algorithms we introduce have two key properties. First, by
preserving the right invariants they guarantee numerical stability. Second, in
closed or periodic systems they do so without degrading the accuracy of an
already-accurate solver.Comment: 41 pages, 10 figure
A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations
A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma
system is presented. The method uses a second or third order discontinuous
Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping
scheme. The method is benchmarked against an analytic solution of a dispersive
electron acoustic square pulse as well as the two-fluid electromagnetic shock
and existing numerical solutions to the GEM challenge magnetic reconnection
problem. The algorithm can be generalized to arbitrary geometries and three
dimensions. An approach to maintaining small gauge errors based on error
propagation is suggested.Comment: 40 pages, 18 figures
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