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7.3 Stability: Criterion for positive coefficients. PC-criterion

Consider the following sum: $$ \begin{align} s=a_1\, x_1+a_2\, x_2+\dots+a_k\, x_k \tag{7.18} \end{align} $$ where \( a_1,a_2,\dots,a_k \) are positive coefficients. Now, by introducing the extrema of \( x_i \) as: $$ \begin{align} x_{min}=\min(x_1,x_2,\dots,x_k) \qquad \text{and} \qquad x_{max}=\max(x_1,x_2,\dots,x_k) \tag{7.19} \end{align} $$

we may deduce from (7.18): $$ \begin{equation} \tag{7.20} x_{min}\cdot (a_1+a_2+\dots+a_k)\leq s\leq x_{max}\cdot (a_1+a_2+\dots+a_k) \end{equation} $$

Note, that the above is valid only when \( a_1,a_2,\dots, a_k \) are all positive. In the following we will consider two cases:

Case 1: \( a_1+a_2+\dots+a_k=1 \)

In this case (7.20) simplifies to: $$ \begin{equation} \tag{7.21} x_{min}\leq s\leq x_{max} \end{equation} $$

Equality in (7.21) is obtained when \( x_1=x_2=\dots=x_k \).

Let us now apply (7.21) on the difference equation in (7.16): $$ \begin{equation*} u^{n+1}_j=D\, (u^n_{j+1}+u^n_{j-1})+(1-2\,D)\, u^n_j \end{equation*} $$

In this case the coefficients are \( a_1=D,\ a_2=D,\ a_3=1-2D \) such that the sum of all coefficients is: \( a_1+a_2+a_3=1 \).

From (7.21) we get: $$ \begin{equation*} \min(u^n_{j+1},u^n_{j},u^n_{j-1})\leq u^{n+1}_j\leq \max(u^n_{j+1},u^n_{j},u^n_{j-1}) \end{equation*} $$

Meaning that \( u^{n+1}_j \) is restricted by the extrema of \( u^{n}_j \), i.e. all the solutions at the previous timestep, and will thus not have to ability to grow without bounds and therefore be stable.

The conditions for stability are that all the coefficients \( a_1,a_2.\dots,a_k \) are positive. As \( D > 0 \), this means that only \( a_3=1-2D \) may become negative. The condition for \( a_3 \) to be positive becomes: \( 1-2D > 0 \) which yields \( D < \frac{1}{2} \). When \( D=\frac{1}{2} \), the coefficient \( a_3=0 \), such that \( a_1+a_2=1 \), which still satisfies the condition. Thus the condition for stability becomes: $$ \begin{align} D \leq \frac{1}{2} \tag{7.22} \end{align} $$

which is commonly referred to as the Bender-Schmidt formula.

For explicit, homogenous schemes which has a constant solution \( u=u_0 \), the sum of the coefficients will often be one. (Substitute e.g. \( u=u_0 \) in (7.16)). This property is due to the form of the difference equations from the Taylor-expansions. (See (forward) , (backward) and (central) differences in (2 Initial value problems for Ordinary Differential Equations)).

Case 2: \( a_1+a_2+\dots+a_k < 1 \)

As a remedy to that the sum of the coefficient do not sum to unity we define \( b=1-(a_1+a_2+\dots + a_k) > 0 \) such that \( a_1+a_2+\dots +a_k+b=1 \).

We may the construct the following sum: \( a=a_1x_1+a_2x_2+\dots+a_k+b\cdot 0 \) which satisfies the conditions for Case 1 and we get: $$ \begin{equation} \tag{7.23} \min(0,x_1,x_2,\dots,x_k)\leq s\leq \max(0,x_1,x_2,\dots,x_k) \end{equation} $$

The only difference from (7.21) being that we have introduced 0 for the \( x \)-es, which naturally has consequences for the extremal values.

Let us look at an example: $$ \begin{equation*} \frac{\partial T}{\partial t}= \alpha \frac{\partial ^2T}{\partial x^2}+bT,\ b=\text{konstant},\ t < t_{maks} \end{equation*} $$

which may be discretized with the FTCS-scheme to yield: $$ \begin{equation*} T^{n+1}_j=D \, (T^n_{j+1}+T^n_{j-1})+(1-2D+\Delta t\cdot b)T^n_j,\qquad D=\alpha \frac{\Delta t}{(\Delta x)^2} \end{equation*} $$

and with the following coefficients \( a_1=a_2=D \) and \( a_3=1-2D+\Delta t\cdot b \) we get: $$ \begin{equation*} a_1+a_2+a_3=1+\Delta t\cdot b\leq 1 \end{equation*} $$ only for negative \( b \)-values.

The condition of positive coefficients then becomes: \( 1-2D+\Delta t\cdot b > 0 \) which corresponds to $$ \begin{align*} 0 < D < \frac{1}{2}+\frac{\Delta t\cdot b}{2} \end{align*} $$ where \( b < 0 \).

In this situation the criterion implies that the \( T-values \) from the difference equation will not increase or be unstable for a negative \( b \). This result agrees well the physics, as a negative \( b \) corresponds to a heat sink.

Note that (7.21) and (7.23) provide limits within which the solution is bounded, and provides a sufficient criteria to prevent the occurrence of unstable oscillations in the solution. This criteria may be far more restrictive that what is necessary for a stable solution. However, in many situations we may be satisfied with such a criteria. The PC-criterion is used frequently on difference equations for which a more exact analysis is difficult to pursue. Note that the PC-criterion may only be applied for explicit schemes if no extra information is provided. For parabolic equations we often have such extra information by means of max/min principles (see (7.5.3 Crank-Nicolson scheme. \( \theta \)-scheme)). Further, the criterion must be modified in case of increasing amplitudes.

One would of course hope for the existence of a necessary and sufficient condition for numerical stability. However, for general difference equations we have no such condition, which is hardly surprising. But a method which often leads to sufficient, and in some cases necessary, conditions for stability, is von Neumann's method. This method involves Fourier-analysis of the linearized difference equation and may be applied for both explicit and implicit numerical schemes. We will present this method in 7.4 Stability analysis with von Neumann's method.