Minimal $ \ell^2 $ norm discrete multiplier method
Copyright policy )Minimal $ \ell^2 $ norm discrete multiplier method
We introduce an extension to the Discrete Multiplier Method (DMM), called Minimal $\ell_2$ Norm Discrete Multiplier Method (MN-DMM), where conservative finite difference schemes for dynamical systems with multiple conserved quantities are constructed procedurally, instead of analytically as in the original DMM. For large dynamical systems with multiple conserved quantities, MN-DMM alleviates difficulties that can arise with the original DMM at constructing conservative schemes which satisfies the discrete multiplier conditions. In particular, MN-DMM utilizes the right Moore-Penrose pseudoinverse of the discrete multiplier matrix to solve an underdetermined least-square problem associated with the discrete multiplier conditions. We prove consistency and conservative properties of the MN-DMM schemes. We also introduce two variants - Mixed MN-DMM and MN-DMM using Singular Value Decomposition - and discuss their usage in practice. Moreover, numerical examples on various problems arising from the mathematical sciences are shown to demonstrate the wide applicability of MN-DMM and its relative ease of implementation compared to the original DMM.
27 pages, 8 figures
- ETH Zurich Switzerland
- University of Northern British Columbia Canada
- University of California, Merced United States
Finite difference and finite volume methods for ordinary differential equations, Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, Numerical Analysis, Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics), Simulation of dynamical systems, Numerical Analysis (math.NA), Dynamical Systems (math.DS), 65L05, 65L12, 65P10, 65Z05, 37M05, 37M15, 37N20, Numerical methods for Hamiltonian systems including symplectic integrators, Numerical methods for initial value problems involving ordinary differential equations, dynamical system, Dynamical Systems, conservative integrator, pseudo inverse, Applications to the sciences, FOS: Mathematics, discrete multiplier method, Schwarzschild geodesic, conserved quantity
Finite difference and finite volume methods for ordinary differential equations, Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, Numerical Analysis, Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics), Simulation of dynamical systems, Numerical Analysis (math.NA), Dynamical Systems (math.DS), 65L05, 65L12, 65P10, 65Z05, 37M05, 37M15, 37N20, Numerical methods for Hamiltonian systems including symplectic integrators, Numerical methods for initial value problems involving ordinary differential equations, dynamical system, Dynamical Systems, conservative integrator, pseudo inverse, Applications to the sciences, FOS: Mathematics, discrete multiplier method, Schwarzschild geodesic, conserved quantity
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