Numerical solution of multiparameter spectral problems by high order finite different schemes

2016 ◽  
Author(s):  
Pierluigi Amodio ◽  
Giuseppina Settanni
Keyword(s):  
Numerical Solution ◽  
High Order ◽  
Spectral Problems

scholarly journals Collocation for High-Order Differential Equations with Lidstone Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Francesco Costabile ◽  
Anna Napoli
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Experiments ◽  
Recurrence Formula ◽  
High Order ◽  
Theoretical Results

A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.

Numerical Solution of Multidimensional Compressible Flow by High Order Nodal Discontinuous Galerkin Method

2013 ◽  
Vol 392 ◽  
pp. 100-104 ◽  
Author(s):  
Fareed Ahmed ◽  
Faheem Ahmed ◽  
Yong Yang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Finite Volume Method ◽  
Discontinuous Galerkin ◽  
Finite Volume ◽  
High Order ◽  
Numerical Predictions ◽  
Volume Method ◽  
Element Method

In this paper we present a robust, high order method for numerical solution of multidimensional compressible inviscid flow equations. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method utilizes the favorable features of Finite Volume Method (FVM) and Finite Element Method (FEM). In this method, space discretization is carried out by finite element discontinuous approximations. The resulting semi discrete differential equations were solved using explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. In this article, we demonstrate the use of exponential filter to remove Gibbs oscillations near the shock waves. Numerical predictions for two dimensional compressible fluid flows are presented here. The solution was obtained with overall order of accuracy of 3. The numerical results obtained are compared with experimental and finite volume method results.

Multi-Integral Method for Solving the Forward Dynamics of Stiff Multibody Systems

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Numerical Solution ◽  
State Vector ◽  
Taylor Series Expansion ◽  
High Order ◽  
The State ◽  
System State ◽  
Time Step ◽  
Entire Interval ◽  
Free Parameters ◽  
Derivatives Of

The Hermite–Obreshkov–Padé (HOP) procedure is an implicit method for the numerical solution of a system of ordinary differential equations (ODEs) applicable to stiff dynamical systems. This procedure applies an Obreshkov condition to multiple derivatives of the system state vector, both at the start and end of a time step in the numerical solution. That condition is shown to be satisfied by the Hermite interpolating polynomial that matches the state vector and its derivatives, also at the start and end of a time step. The Hermite polynomial, in turn, can be specified in terms of the system state and its derivatives at the start of a step together with a collection of free parameters. Adjusting these free parameters to minimize magnitudes of the ODE residual and its derivatives at the end of a step serves as a proxy for matching the system state and its derivatives. A high-order Taylor expansion at the start of a time step interval models the residual and its derivatives over the entire interval. A variant of this procedure adjusts those parameters to match integrals of the system state over the duration of that interval. This is done by minimizing magnitudes of integrals of the ODE residual calculated from the extrapolating Taylor-series expansion, a process that avoids the need to determine integration constants for multiple integrals of the state. This alternative method eliminates the calculation of high-order derivatives of the system state and hence avoids loss in accuracy from floating-point round off. Numerical performance is evaluated on a dynamically unbalanced constant-velocity (CV) coupling having a high spring rate constraining shaft deflection.

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