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An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems

NUMERICAL ALGORITHMS 2020年第3期84卷 1121-1153页

作者： Baccouch, Mahboub Univ Nebraska Dept Math Omaha NE 68182 USA

In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the formu ''=f(x,u),x is an element of[a,b subject to some suitable boundary conditions at the endpoint of the interval [a,b]. We first use the superconvergence results proved in the first part of this paper as reported by Baccouch (Numer. Algorithm.79(3), 697-7182018) to show that the significant parts of the local discretization errors are proportional to (p+ 1)-degree Radau polynomials, when polynomials of total degree not exceedingpare used. These new results allow us to construct a residual-baseda posteriorierror estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed aposteriorierror estimates converge to the exact errors in theL(2)-norm with order of convergencep+ 3/2. Finally, we present a local AMR procedure that makes use of our local and globala posteriorierror estimates. Our proofs are valid for arbitrary regular meshes and forP(p)polynomials withp >= 1. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.

关键词： nonlinear two-point boundary-value problems Local discontinuous Galerkin method Superconvergence A posteriorierror estimates Adaptive mesh refinement

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A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems

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NUMERICAL ALGORITHMS 2018年第3期79卷 697-718页

作者： Baccouch, Mahboub Univ Nebraska Dept Math Omaha NE 68182 USA

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u = f (t, u), which arise in a wide variety of engineering applications. We prove the L-2 stability of the LDG scheme and optimal L-2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p +1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p +1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p +3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p +2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p 1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.

关键词： nonlinear two-point boundary-value problems Local discontinuous Galerkin method A priori error estimates Superconvergence Gauss-Radau projections L-2 stability 65L10 65L20 65L60 65L70

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DECOMPOSITION TECHNIQUE AND OPTIMAL TRAJECTORIES FOR THE AEROASSISTED FLIGHT EXPERIMENT

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1991年第2期69卷 201-234页

作者： MIELE, A WANG, T DEATON, AW NASA GEORGE C MARSHALL SPACE FLIGHT CTRSYST ANAL & INTEGRAT LABFLIGHT MECH BRANCHHUNTSVILLEAL 35812

The aeroassisted flight experiment (AFE) refers to a spacecraft to be launched and then recovered by the space shuttle in 1994. It simulates a transfer from a geosynchronous Earth orbit (GEO) to a low Earth orbit (LEO). Specifically, the AFE spacecraft is released from the space shuttle and is accelerated by means of a solid rocket motor toward Earth, so as to achieve atmospheric entry conditions close to those of a spacecraft returning from GEO. Following the atmospheric pass, the AFE spacecraft ascends to the specified LEO via an intermediate parking Earth orbit (PEO). The final maneuver includes the rendezvous with and the capture by the space shuttle. The entry and exit orbital planes of the AFE spacecraft are identical with the orbital plane of the space shuttle. In this paper, with reference to the AFE spacecraft, an actual GEO-to-LEO transfer is considered and optimal trajectories are determined by minimizing the total characteristic velocity. The optimization is performed with respect to the time history of the controls (angle of attack and angle of bank), the entry path inclination and the flight time being free. two transfer maneuvers are considered: (DA) direct ascent to LEO;(IA) indirect ascent to LEO via PEO. While the motion of the AFE spacecraft in a 3D-space is described by a system of six ODEs, substantial simplifications are possible if one exploits these facts: (i) the instantaneous orbital plane is nearly identical with the initial orbital plane;(ii) the bank angle is small;and (iii) the Earth's angular velocity is relatively small. Under these assumptions, the complete system can be decoupled into two subsystems, one describing the longitudinal motion and one describing the lateral motion. The angle of attack history, the entry path inclination, and the flight time are determined via the longitudinal motion subsystem;in this subsystem, the total characteristic velocity is minimized subject to the specified LEO requirement. The angle of bank hist

关键词： Flight mechanics astrodynamics hypervelocity flight aeroassisted orbital transfer aeroassisted flight experiment optimal trajectories guidance trajectories decomposition techniques longitudinal motion lateral motion sequential gradient-restoration algorithm nonlinear two-point boundary-value problems

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A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems

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Journal of Computational and Applied Mathematics 1988年第1期23卷 1-10页

作者： Chawla, M.M. Subramanian, R. Department of Mathematics Indian Institute of Technology New Delhi 110016 Hauz Khas India

Bickley [5] had suggested the use of cubic splines for the solution of general linear two-point boundary-value problems. It is well known since then that this method gives only order h² uniformly convergent approximations. But cubic spline interpolation itself is a fourth-order process. We present a new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems: y″ = f(x, y, y′), a < x < b, α₀y(a) - α₀y′(a) = A, β₀y(b) + β₁y′(b) = B. We generate the solution at the nodal points by a fourth-order method and then use 'conditions of continuity' to obtain smoothed approximations for the second derivatives of the solution needed for the construction of the cubic spline solution. We show that our method provides order h⁴ uniformly convergent approximations over [a, b]. The fourth order of the presented method is demonstrated computationally by two examples. © 1988.

关键词： Cubic spline finite differences interpolation nonlinear two-point boundary-value problems

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PARALLEL ALGORITHMS FOR SOLVING NON-LINEAR 2-point boundary-value-problems WHICH ARISE IN OPTIMAL-CONTROL

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1980年第1期30卷 53-71页

作者： TRAVASSOS, R KAUFMAN, H RENSSELAER POLYTECH INST DEPT ELECT & SYST ENGNTROYNY 12181

This paper describes a collection of parallel optimal control algorithms which are suitable for implementation on an advanced computer with the facility for large-scale parallel processing. Specifically, a parallel nongradient algorithm and a parallel variablemetric algorithm are used to search for the initial costate vector that defines the solution to the optimal control problem. To avoid the computational problems sometimes associated with simultaneous forward integration of both the state and costate equations, a parallel shooting procedure based upon partitioning of the integration interval is considered. To further speed computations, parallel integration methods are proposed. Application of this all-parallel procedure to a forced Van der Pol system indicates that convergence time is significantly less than that required by highly efficient serial procedures.

关键词： Parallel processing nonlinear two-point boundary-value problems optimal control parallel shooting parallel integration parallel minimization algorithms

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