4.1. Concepts and definitions#
4.1.1. Boundedness#
Definition
A function \(f\) defined on a set \(X\) with real or complex values is called bounded if there exists a real number \(M\) such that
for all \(x\) in \(X\).
The function \(f(x)=sin(x)\) is an example of a bounded function, since \(|sin(x)|\leq 1\) for all values of \(x\).
A uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant.
4.1.2. Monotonicity#
Definition
A scheme is monotonous if
\( q_m^n\geq q_{m+1}^n\,\, \forall\, m \).
Could you think of any functions that fullfuill this criteria?
4.1.3. Error of a numerical approximation#
First, let’s clarify the expression “converge to the solution of the exact equation”, by defining the error of the numerical approximation.
Definition
The error of the numerical approximation \(q_m^n\), \(\epsilon_m^n\), is
where \(q_m^n\) is the numerical solution and \(q(m\Delta x,n\Delta t)\) is the exact solution at \((x_m,t^n)\).
We can now define more precisely the concepts of boundedness and convergence. Let us consider the following questions:
What happens to \(\epsilon_m^n\) as \(\Delta t, \Delta x \to 0\) for fixed \(n\Delta t\)?
What happens to \(\epsilon_m^n\) as \(n\to\infty\) for fixed \(\Delta t\) and \(\Delta x\)?
The first questions leads us to the concepts of consistency and convergence, and the second question leads us to the concept of numerical stability.
4.1.4. Consistency#
Definition
A difference equation is consistent when it approaches the corresponding PDE as \(\Delta t, \Delta x \to 0\).
The truncation error of the leapfrog scheme applied to the linear advection equation is:
or \(Tr = O(\Delta t^2)+O(\Delta x^2)\), which clearly converges to zero as \(\Delta t, \Delta x \to 0\). Thus the leapfrog discretization of the linear advection equation is consistent.
Important
Numerical schemes with truncation errors with terms \(O(\Delta t^p)\) or \(O(\Delta x^p)\) are consistent if \(p \ge 1\).
4.1.5. Convergence#
Having a concistent numerical scheme is not a guarantee that we will obtain meaningful results. We must also demand that, over an interval \(n\Delta t\), \(\epsilon_m^n \to 0\) as \(\Delta t, \Delta x \to 0\).
Definition
A numerical solution is convergent if, for fixed \(n\Delta t\), \(\epsilon_m^n \to 0\) as \(\Delta t, \Delta x \to 0\).
If a numerical scheme gives convergent solutions for any initial condition, the numerical scheme is said to be convergent.
4.1.6. Numerical stability#
If \(q(m\Delta x,n\Delta t)\) is bounded, we can expect \(\epsilon_m^n\) to remain bounded if \(q_m^n\) remains bounded.
This leads us to the definition of stability of a numerical scheme:
Definition
A finite-difference scheme is stable if its solutions remain uniformly bounded functions of the initial condition, for all sufficiently small \(\Delta t\). \(||q^n||\leq C||q^0||\,\,\,\,\forall\,\, n\Delta t\leq T\)
A more strict definition of stability is set by choosing \(C=1\) in the definition above: $\( ||q^n||\leq ||q^0||\,\,\,\,\forall\,\, n\Delta t\leq T \)$ (eq:StrictStability)
Important
The concept of stability is not directly related to the underlying PDE.
Convergence is not so easily demonstrated as it involves the exact equation, whose solution may not be known. Stability is easier to demonstrate as it involved the finite-difference scheme. Luckily, we can infer convergence from stability in certain cases using the Lax-Richtmyer Equivalence Theorem [CRB11].
Lax-Richtmyer Equivalence Theorem
Given a properly posed, linear initial value problem and a finite-difference approximation that satisfies the consistency condition, stability is the necessary and sufficient condition for convergence.
Therefore, if we can show consistency and stability, the convergence of the finite-difference scheme is guaranteed by the Lax Equivalence Theorem.