Convergence Of High-order Methods Research Articles

On the Semi-Local Convergence of a Noor–Waseem-like Method for Nonlinear Equations

The significant feature of this paper is that the semi-local convergence of high order methods for solving nonlinear equations defined on abstract spaces has not been studied extensively as done for the local convergence by a plethora of authors which is certainly a more interesting case. A process is developed based on majorizing sequences and the notion of restricted Lipschitz condition to provide a semi-local convergence analysis for the third convergent order Noor–Waseem method. Due to the generality of our technique, it can be used on other high order methods. The convergence analysis is enhanced. Numerical applications complete are used to test the convergence criteria.

On the local convergence of higher order methods in Banach spaces

On the local convergence of higher order methods in Banach spaces

Rate of convergence of higher order methods

Methods like the Chebyshev and the Halley method are well known methods for solving nonlinear systems of equations. They are members in the Halley class of methods and all members in this class have local and third order rate of convergence. They are single point iterative methods using the first and second derivatives. Schröderʼs method is another single point method using the first and second derivatives. However, this method is only quadratically convergent. In this paper we derive a unified framework for these methods and show their local convergence and rate of convergence. We also use the same approach to derive inexact methods. The methods in the Halley class require solution of two linear systems of equations for each iteration. However, in the Chebyshev method the coefficient matrices will be the same. Using the unified framework we show how to extend this to all methods in the class. We will illustrate these results with some numerical experiments.

A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids

The convergence of high-order methods, such as recently developed spectral difference (SD) method, can be accelerated using both implicit temporal advancement and a p-multigrid ( p = polynomial degree) approach. A p-multigrid method is investigated in this paper for solving SD formulations of the scalar wave and Euler equations on unstructured grids. A fast preconditioned lower–upper symmetric Gauss–Seidel (LU-SGS) relaxation method is implemented as an iterative smoother. Meanwhile, a Runge–Kutta explicit method is employed for comparison. The multigrid method considered here is nonlinear and utilizes full approximation storage (FAS) [Ta’asan S. Multigrid one-shot methods and design strategy, Von Karman Institute Lecture Note, 1997 [28]] scheme. For some p-multigrid calculations, blending implicit and explicit smoothers for different p-levels is also studied. The p-multigrid method is firstly validated by solving both linear and nonlinear 2D wave equations. Then the same idea is extended to 2D nonlinear Euler equations. Generally speaking, we are able to achieve speedups of up to two orders using the p-multigrid method with the implicit smoother.

Grid convergence of high order methods for multiscale complex unsteady viscous compressible flows

Grid convergence of several high order methods for the computation of rapidly developing complex unsteady viscous compressible flows with a wide range of physical scales is studied. The recently developed adaptive numerical dissipation control high order methods referred to as the ACM and wavelet filter schemes are compared with a fifth-order weighted ENO (WENO) scheme. The two 2-D compressible full Navier–Stokes models considered do not possess known analytical and experimental data. Fine grid solutions from a standard second-order TVD scheme and a MUSCL scheme with limiters are used as reference solutions. The first model is a 2-D viscous analog of a shock tube problem which involves complex shock/shear/boundary-layer interactions. The second model is a supersonic reactive flow concerning fuel breakup. The fuel mixing involves circular hydrogen bubbles in air interacting with a planar moving shock wave. Both models contain fine scale structures and are stiff in the sense that even though the unsteadiness of the flows are rapidly developing, extreme grid refinement and time step restrictions are needed to resolve all the flow scales as well as the chemical reaction scales. Our computations were all made on uniform grids, and our conclusions cannot be directly carried over to, for example, curvilinear grids.