2. Numerical Schemes 2¶

2.1. Definition of Consistency¶

Consistency is a condition on the numerical scheme, i.e.
The scheme must tend to the differential equation when the step in time and space tend to zero.

2.2. Definition of Stability¶

Stability is a condition on the numerical solution, i.e.
All errors must remain bounded when the iteration process progresses
For finite values of \(\Delta t\) and \(\Delta x\), the error has to remain bounded when the number of time steps \(n\) tends to infinity.
If the error:

\[\bar{\epsilon}_i^n = u_i^n-\bar{u}_i^n\]

where:

\(\qquad u_i^n\) = computed solution

\(\qquad \bar{u}_i^n\) = discretised solution

Then the stability condition is:

\[\lim_{n \to \infty} \left \vert \bar{\epsilon}_i^n \right \vert \le \kappa \quad \text{ at fixed } \Delta t\]

NOTE:

The stability condition is a requirement on the numerical scheme only (does not require any condition on the differential equation)
Stability does not ensure that the error will not become unacceptably large at intermediate timesteps \(t^n = n \Delta t\) (while remaining bounded)
A more general definition of stability later!

2.3. Definition of Convergence¶

Convergence is a condition on the numerical solution
The numerical solution must tend to the exact solution of the mathematical model, when steps in \(t\) and \(x\) tend to zero (i.e. when the mesh is refined).
If the error:

\[\tilde{\epsilon}_i^n = u_i^n-\tilde{u}_i^n\]

where:

\(\qquad u_i^n\) = computed solution

\(\qquad \tilde{u}_i^n = \tilde{u}_i^n(i\Delta x, n \Delta t)\) = solution of differential equation

Convergence condition:

\[\lim_{\Delta x \to 0, \Delta t \to 0} \left \vert \tilde{\epsilon}_i^n \right \vert = 0\]

2.4. Equivalence Theorem of Lax¶

For a well-posed IVP and a consistent discretisation scheme, stability is the necessary and sufficient condition for convergence

_images/consistency_stability_convergence.png

2.5. Consistency and the modified differential equation¶

A consistent scheme is one in which the truncation error tends to zero for \(\Delta t\), \(\Delta x \rightarrow 0\)
e.g. CD in space and FD in time for linear convection:

(1)\[{{u_i^{n+1} - u_i^n} \over {\Delta t}} + c {{u_{i+1}^n - u_{i-1}^n} \over {2 \Delta x}}=0\]

Taylor expansions:

(2)\[u_i^{n+1} = u_i^n + \Delta t \left . {\partial u \over \partial t} \right \vert_i^n + {\Delta t^2 \over 2} \left . {\partial^2 u \over \partial t^2} \right \vert_i^n + {\Delta t^3 \over 6} \left . {\partial^3 u \over \partial t^3} \right \vert_i^n + {\Delta t^4 \over 24} \left . {\partial^4 u \over \partial t^4} \right \vert_i^n + {\Delta t^5 \over 120} \left . {\partial^5 u \over \partial t^5} \right \vert_i^n\]

(3)\[u_{i+1}^n = u_i^n + \Delta x \left . {\partial u \over \partial x} \right \vert_i^n + {\Delta x^2 \over 2} \left . {\partial^2 u \over \partial x^2} \right \vert_i^n + {\Delta x^3 \over 6} \left . {\partial^3 u \over \partial x^3} \right \vert_i^n + {\Delta x^4 \over 24} \left . {\partial^4 u \over \partial x^4} \right \vert_i^n + {\Delta x^5 \over 120} \left . {\partial^5 u \over \partial x^5} \right \vert_i^n\]

(4)\[u_{i-1}^n = u_i^n - \Delta x \left . {\partial u \over \partial x} \right \vert_i^n + {\Delta x^2 \over 2} \left . {\partial^2 u \over \partial x^2} \right \vert_i^n - {\Delta x^3 \over 6} \left . {\partial^3 u \over \partial x^3} \right \vert_i^n + {\Delta x^4 \over 24} \left . {\partial^4 u \over \partial x^4} \right \vert_i^n - {\Delta x^5 \over 120} \left . {\partial^5 u \over \partial x^5} \right \vert_i^n\]

Re-arranging Equation (2):

(5)\[{{u_i^{n+1} - u_i^n} \over {\Delta t}} = \left . {\partial u \over \partial t} \right \vert_i^n + {\Delta t \over 2} \left . {\partial^2 u \over \partial t^2} \right \vert_i^n + O(\Delta t^2)\]

Equation (3) minus Equation (4) and rearranging:

(6)\[{c \over {2 \Delta x}} {(u_{i+1}^n - u_{i-1}^n)} = c \left . {\partial u \over \partial x} \right \vert_i^n + c {\Delta x^2 \over 6} \left . {\partial^3 u \over \partial x^3} \right \vert_i^n + O(\Delta x^4)\]

Including the truncation error in (1), defined as the difference between the numerical approximation and the differential equation:

\[{{u_i^{n+1} - u_i^n} \over {\Delta t}} + c {{u_{i+1}^n - u_{i-1}^n} \over {2 \Delta x}} - \left ( \left . {\partial u \over \partial t} \right \vert_i^n + \left . c {\partial u \over \partial x} \right \vert_i^n \right ) = {\Delta t \over 2} \left . {\partial^2 u \over \partial t^2} \right \vert_i^n + c {\Delta x^2 \over 6} \left . {\partial^3 u \over \partial x^3} \right \vert_i^n + O(\Delta t^2) + O(\Delta x^4)\]

The RHS of the above equation is the truncation error

\[\epsilon_T = {\Delta t \over 2} \left . {\partial^2 u \over \partial t^2} \right \vert_i^n + c {\Delta x^2 \over 6} \left . {\partial^3 u \over \partial x^3} \right \vert_i^n + O(\Delta t^2) + O(\Delta x^4)\]

The scheme is consistent because when \(\Delta t\), \(\Delta x \rightarrow 0\), \(\epsilon_T \rightarrow 0\)

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