Construction of Higher-Order Accurate Time-Step Integration Algorithms by Equal-Order Polynomial Projection

2005 ◽  
Vol 11 (1) ◽  
pp. 19-49 ◽  
Author(s):  
T. C. Fung
Keyword(s):  
Integral Relation ◽  
Higher Order ◽  
Numerical Results ◽  
Numerical Dissipation ◽  
Time Interval ◽  
Time Step ◽  
Higher Order Terms ◽  
Optimal Polynomial ◽  
Integration Algorithms ◽  
New Framework

This paper presents a new framework to construct higher-order accurate time-step integration algorithms based on weakly enforcing the differential/integral relation. The dependent variable and its time derivatives are assumed to be polynomials of equal order. A differential equation is then transformed into an algebraic equation directly. The main issue is how to approximate the integral of a polynomial by another polynomial of the same degree. Various methods to determine the optimal representation (or projection) are considered. It is shown that to reproduce numerical results equivalent to the Padé or generalized Padé approximations, the coefficients of the optimal polynomial representation are related to the weighting parameters derived previously for time-step integration algorithms with predetermined coefficients. A special feature for the present formulation is that the same procedure can be used to solve first-, second-, and even higher-order non-homogeneous initial value problems in a unified manner. The resultant algorithms are higher-order accurate and unconditionally A-stable with controllable numerical dissipation. It is also shown that for the numerical results to maintain higher-order accuracy at the end of a time interval, the higher-order terms in the excitation have to be projected as polynomials of lower degree within the present framework as well. Numerical examples are given to illustrate the validity of the present formulations.

scholarly journals COMPUTING THE ASYMPTOTIC EXPANSION OF THE MEDIAN OF THE ERLANG DISTRIBUTION

2012 ◽  
Vol 17 (2) ◽  
pp. 281-292 ◽  
Author(s):  
Pedro Jodr´a
Keyword(s):  
Asymptotic Expansion ◽  
Higher Order ◽  
Numerical Results ◽  
Complete Monotonicity ◽  
Partial Sums ◽  
Erlang Distribution ◽  
Higher Order Terms ◽  
Erlang Distributions

We describe an algorithm for computing a large number of coefficients in the asymptotic expansion of the median of the Erlang distribution. In particular, in this paper we present the values of the first sixty coefficients which allow us to assess the importance of the higher-order terms in the behavior of the partial sums of that asymptotic expansion. As a consequence, we provide tight bounds for the median of the Erlang distribution and we also see that a conjecture concerning the complete monotonicity of a sequence of medians of the Erlang distributions is supported by numerical results.

Effective Higher-Order Time Integration Algorithms for the Analysis of Linear Structural Dynamics

2017 ◽  
Vol 84 (7) ◽  
Author(s):  
Wooram Kim ◽  
J. N. Reddy
Keyword(s):  
Structural Dynamics ◽  
Displacement Vector ◽  
Time Integration ◽  
Higher Order ◽  
Time Interval ◽  
Weighted Residual Method ◽  
Residual Vector ◽  
Integral Forms ◽  
Integration Algorithms ◽  
Hermite Approximation

For the development of a new family of higher-order time integration algorithms for structural dynamics problems, the displacement vector is approximated over a typical time interval using the pth-degree Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted-residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted-residual statements in algebraic forms, and then, these parameters are optimized by using the single-degree-of-freedom problem and its exact solution to achieve improved accuracy and unconditional stability. As a result of the pth-degree Hermite approximation of the displacement vector, pth-order (for dissipative cases) and (p + 1)st-order (for the nondissipative case) accurate algorithms with dissipation control capabilities are obtained. Numerical examples are used to illustrate performances of the newly developed algorithms.

scholarly journals Hamiltonian reduction of Vlasov–Maxwell to a dark slow manifold

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
George Miloshevich ◽  
Joshua W. Burby
Keyword(s):  
Hamiltonian Structure ◽  
Slow Manifold ◽  
Higher Order ◽  
Hamiltonian Reduction ◽  
Time Interval ◽  
Maxwell System ◽  
Infinite Dimensional ◽  
Light Waves ◽  
Higher Order Terms ◽  
Higher Order Corrections

We show that non-relativistic scaling of the collisionless Vlasov–Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov–Maxwell phase space. Vlasov–Maxwell dynamics restricted to the slow manifold recovers the Vlasov–Poisson and Vlasov–Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov–Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov–Maxwell solutions that do not excite light waves, and are therefore ‘dark’. We provide a heuristic lower bound for the time interval over which Vlasov–Maxwell solutions initialized optimally near the slow manifold remain dark. We also show how the dynamics on the slow manifold naturally inherits a Hamiltonian structure from the underlying system. After expressing this structure in a simple form, we use it to identify a manifestly Hamiltonian correction to the Vlasov–Darwin model. The derivation of higher-order terms is reduced to computing the corrections of the system Hamiltonian restricted to the slow manifold.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

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