Contents
How are convergence rates of finite difference schemes determined?
The consistency and the stability of the schemes are described. By the support of the numerical problems convergence rates of the schemes have been determined. It is found that both methods are first order accurate in the spatial dimension in – norm.
What are the results of the finite difference method?
Theoretical results have been obtained regarding the accuracy, stability and convergence of the finite difference methods (FDMs) for solving this equation.
How are convergence rates of FTCS and BTCS schemes determined?
In this work, it is aimed to determine the convergence rates of the FTCS and BTCS schemes for solving equations (1.1a) – (1.1d) which are often encountered in engineering applications. The Neumann boundary condition specifies the temperature gradients across the boundaries as well as the initial temperature distribution within the rod.
Is the Lax equivalence theorem necessary for convergence?
Since convergence is difficult to prove directly, we use an equivalent result known as the Lax Equivalence Theorem which stated that, for a given properly posed linear consistent finite difference approximation to Partial differential equation (PDE), stability is necessary and sufficient for convergence [3].
Can a grid be in the asymptotic range of convergence?
The order of accuracy of the boundary conditions can be one order of accuracy lower than the interior order of accuracy without degrading the overall global accuracy. Assessing the accuracy of code and caluculations requires that the grid is sufficiently refined such that the solution is in the asymptotic range of convergence.
Is the FTCS method stable with Dirichlet boundary?
In [1, 3, 5], it is stated that for any time step size in the time range [0, T] and for space step size , FTCS method is stable if (r is stability limit) and BTCS method is unconditionally stable with Dirichlet boundary conditions. The authors also stated that, these methods are first-order accurate in time and second-order accurate in space.