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The Scarborough criterion is used for satisfying convergence of a solution while solving linear equations using an iterative method.

Introduction

The Scarborough criterion formulated by Scarborough (1958), can be expressed in terms of the values of the coefficients of the discretised equations:[1][2]

$$\frac{\sum |a_{nb}|}{|a'_{p}|} \begin{cases} \leq 1, & \text{at all nodes}\\ <1, & \text{at one node at least} \end{cases}$$

Here a'p is the net coefficient of the central node P and the summation in the numerator is taken over all the neighbouring nodes.

This is a sufficient condition, not a necessary one. This means that we can get convergence,even the equations violates the criterion.[3]
The satisfaction of this criterion ensures that the equations will be converged by at least one iterative method.[3]

Gauss–Seidel method

If Scarborough criterion is not satisfied then Gauss–Seidel method iterative procedure is not guaranteed to converge a solution. This criterion is a sufficient condition,[3] not a necessary one. If this criterion is satisfied then it means equation will be converged by at least one iterative method. The Scarborough criterion is used as a sufficient condition for convergent iterative method. The finite volume method uses this criterion for obtaining a convergent solution and implementing boundary conditions.
Diagonal dominance

If the differencing scheme produces coefficients that satisfy the above criterion the resulting matrix of coefficients is diagonally dominant.[4] To achieve diagonal dominance we need large values of net coefficient so the linearisation practice of source terms should ensure that SP is always negative. If this is the case –SP is always positive and adds to aP. Diagonal dominance is a desirable feature for satisfying the boundedness criterion. This states that in the absence of sources the internal nodal values of the property ф should be bounded by its boundary values. Hence in a steady state conduction problem without sources and with boundary temperatures of 500 °C and 200 °C all interior values of T should be less than 500 °C and greater than 200 °C.[2]

Computational fluid dynamics
Linear equation

References

James Blaine Scarborough (1955). Numerical Mathematical Analysis. Johns Hopkins Press.
Henk Kaarle Versteeg; Weeratunge Malalasekera (1 January 2007). An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education Limited. ISBN 978-0-13-127498-3.
Suhas Patankar (1 January 1980). Numerical Heat Transfer and Fluid Flow. CRC Press. pp. 64–. ISBN 978-0-89116-522-4.

W. J. Minkowycz (28 March 1988). Handbook of Numerical Heat Transfer. Wiley. ISBN 978-0-471-83093-1.

Introduction to Computational Fluid Dynamics and Principles of Conservation - video lecture
Overview of Numerical Methods
Implementation of BC in FVM

1 Examples of spectral methods
1.1 A concrete, linear example
1.1.1 Algorithm
1.2 A concrete, nonlinear example
2 A relationship with the spectral element method