Based on the rule that numerical algorithms should preserve the intrinsic properties of the original problem as many as possible, we propose two local energy-preservingalgorithms for the nonlinear fourth-order Schrod...
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Based on the rule that numerical algorithms should preserve the intrinsic properties of the original problem as many as possible, we propose two local energy-preservingalgorithms for the nonlinear fourth-order Schrodinger equation with a trapped term. The local energy conservation law is preserved on any local time-space region. With appropriate boundary conditions, the first algorithm will be both globally charge- and energy-preserving and the second one will be energy-preserving. Numerical experiments show that the proposed algorithms provide more accurate solution than many existing methods and also exhibit excellent performance in preserving conservation laws. (C) 2016 Elsevier Inc. All rights reserved.
Exponential Runge-Kutta (ERK) and partitioned exponential Runge-Kutta (PERK) methods are developed for solving initial value problems with vector fields that can be split into conservative and linear nonconservative p...
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Exponential Runge-Kutta (ERK) and partitioned exponential Runge-Kutta (PERK) methods are developed for solving initial value problems with vector fields that can be split into conservative and linear nonconservative parts. The focus is on linearly damped ordinary differential equations that possess certain invariants when the damping coefficient is zero, but, in the presence of constant or time-dependent linear damping, the invariants satisfy linear differential equations. Similar to the way that Runge-Kutta and partitioned Runge-Kutta methods preserve quadratic invariants and symplecticity for Hamiltonian systems, ERK and PERK methods exactly preserve conformal symplecticity, as well as decay (or growth) rates in linear and quadratic invariants, under certain constraints on their coefficient functions. Numerical experiments illustrate the higher-order accuracy and structure-preserving properties of various ERK methods, demonstrating clear advantages over classical conservative Runge-Kutta methods, as well as usefulness for solving a wide range of differential equations.
Past work on integration methods that preserve a conformal symplectic structure focuses on Hamiltonian systems with weak linear damping. In this work, systems of PDEs that have conformal symplectic structure in time a...
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Past work on integration methods that preserve a conformal symplectic structure focuses on Hamiltonian systems with weak linear damping. In this work, systems of PDEs that have conformal symplectic structure in time and space are considered, meaning conformal symplecticity is fully generalized for PDEs. Using multiple examples, it is shown that PDEs with this particular structure have interesting applications. What it means to preserve a multi-conformal-symplectic conseivation law numerically is explained, along with presentation of two numerical methods that preserve such properties. Then, the advantages of the methods are briefly explored through applications to linear equations, consideration of momentum and energy dissipation, and backward error analysis. Numerical simulations for two PDEs illustrate the properties of the methods, as well as the advantages over other standard methods. (C) 2017 Elsevier B.V. All rights reserved.
A numerical algorithm for solving full gyro orbit of relativistic charged particle motion in magnetized plasmas is presented. The algorithm developed here achieves the following features simultaneously. (1) The time a...
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A numerical algorithm for solving full gyro orbit of relativistic charged particle motion in magnetized plasmas is presented. The algorithm developed here achieves the following features simultaneously. (1) The time advancement is explicit, and (2) the integration is performed with respect to the observation time in a laboratory frame. (3) It is suitable for accurate long time integration with its volume preserving properties in the phase space. (4) The algorithm can properly treat the E x B drift velocity in electromagnetic fields for large relativistic factors, and (5) can be extended to arbitrary high orders with the aid of symmetric composition methods. Because our algorithm is formulated in the Lorentz covariant form, explicit conservation of the Minkowski norm is not assumed. Nevertheless, no secular growth of the numerical errors in the norm does occur, and its influence can be minimized up to the levels of round-off errors when a high-order scheme is applied. Numerical results are compared with explicit and implicit Runge-Kutta methods. The numerical accuracy and the computational efficiency are discussed for long time integration in toroidal magnetic field configuration. (C) 2017 Elsevier B.V. All rights reserved.
The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In...
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The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative （dissipative） nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.
In this paper, combining the ideas of exponential integrators and discrete gradients, we propose and analyze a new structure-preserving exponential scheme for the conservative or dissipative system (y) over dot = Q(My...
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In this paper, combining the ideas of exponential integrators and discrete gradients, we propose and analyze a new structure-preserving exponential scheme for the conservative or dissipative system (y) over dot = Q(My vertical bar del U(y)), where Q is a dxd skew-symmetric or negative semidefinite real matrix, M is a dxd symmetric real matrix, and U : R-d -> R is a differentiable function. We present two properties of the new scheme. The paper is accompanied by numerical results that demonstrate the remarkable superiority of our new scheme in comparison with other structure-preserving schemes in the scientific literature.
Numerical methods for solving linearly damped Hamiltonian systems are constructed using the popular Stormer-Verlet and implicit midpoint methods. Each method is shown to preserve dissipation of symplecticity and dissi...
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Numerical methods for solving linearly damped Hamiltonian systems are constructed using the popular Stormer-Verlet and implicit midpoint methods. Each method is shown to preserve dissipation of symplecticity and dissipation of angular momentum of an N-body system with pairwise distance dependent interactions. Necessary and sufficient conditions for second order accuracy are derived. Analysis for linear equations gives explicit relationships between the damping parameter and the step size to reveal when the methods are most advantageous;essentially, the damping rate of the numerical solution is exactly preserved under these conditions. The methods are applied to several model problems, both ODEs and PDEs. Additional structure preservation is discovered for the discretized PDEs, in one case dissipation in total linear momentum and in another dissipation in mass are preserved by the methods. The numerical results, along with comparisons to standard Runge-Kutta methods and another structure-preserving method, demonstrate the usefulness and strengths of the methods.
Claims were made in an article by Wang and Ma in 2013 that they had devised an algorithm for the quaternion LU decomposition that was significantly faster than the LU decomposition implemented in the Quaternion Toolbo...
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Claims were made in an article by Wang and Ma in 2013 that they had devised an algorithm for the quaternion LU decomposition that was significantly faster than the LU decomposition implemented in the Quaternion Toolbox for MATLAB (QTFM). These claims have been tested and found to be unsupported by MATLAB code supplied to the author by Wang and Ma. The author's tests are presented, and test code made available as supplementary material. It is found that not only is the QTFM code faster, but that Wang and Ma's algorithm has run-time that scales with the square of the size of the matrix, whereas the algorithm in QTFM has run time approximately linear in matrix size. These findings are consistent with an inspection of the code. (C) 2014 Elsevier B.V. All rights reserved.
When the matrices $A$ and $Q$ have special structure, the structure-preservingalgorithm was used to compute the stabilizing solution of the complex matrix equation $X+A^TX^{-1}A=Q.$ In this paper, we study the numeric...
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When the matrices $A$ and $Q$ have special structure, the structure-preservingalgorithm was used to compute the stabilizing solution of the complex matrix equation $X+A^TX^{-1}A=Q.$ In this paper, we study the numerical methods to solve the complex
symmetric stabilizing solution of the general matrix equation $X+A^TX^{-1}A=Q.$ We
not only establish the global convergence for the methods under an assumption, but
also show the feasibility and effectiveness of them by numerical experiments.
Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. Continuation MPC, suggested by T. Ohtsuka ...
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Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. Continuation MPC, suggested by T. Ohtsuka in 2004, uses Krylov-Newton iterations. Continuation MPC is suitable for nonlinear problems and has been recently adopted for minimum time problems. We extend the continuation MPC approach to a case where the state is implicitly constrained to a smooth manifold. We propose an algorithm for on-line controller implementation and demonstrate its numerical effectiveness for a test problem on a hemisphere.
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