Convergence Rates of Iterative Treatments of Partial Differential Equations

1950 ◽  
Vol 4 (30) ◽  
pp. 65 ◽  
Author(s):  
Stanley P. Frankel
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Partial Differential

Computations of Higher Class Solutions of Partial Differential Equations

2001 ◽  
Author(s):  
K. S. Surana ◽  
M. A. Bona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Integral Form ◽  
Mathematical Framework ◽  
Partial Differential ◽  
Computational Framework ◽  
Fundamental Point ◽  
Adaptive Processes ◽  
Fixed Order

Abstract This paper presents a new computational strategy, computational framework and mathematical framework for numerical computations of higher class solutions of differential and partial differential equations. The approach presented here utilizes ‘strong forms’ of the governing differential equations (GDE’s) and least squares approach in constructing the integral form. The conventional, or currently used, approaches seek the convergence of a solution in a fixed (order) space by h, p or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek convergence of the computed solution by changing the orders of the spaces of the basis functions. With this approach convergence rates much higher than those from h,p–processes are achievable and the progressively computed solutions converge to the ‘strong’ i.e. ‘theoretical’ solutions of the GDE’s. Many other benefits of this approach are discussed and demonstrated. Stationary and time-dependant convection-diffusion and Burgers equations are used as model problems.

scholarly journals A Sixth Order Accuracy Solution to a System of Nonlinear Differential Equations with Coupled Compact Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Don Liu ◽  
Qin Chen ◽  
Yifan Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Sixth Order ◽  
Compact Schemes ◽  
Order Convergence ◽  
Order Accuracy ◽  
Partial Differential

A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates.

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