where y, is the value of y calculated by the method after n steps, and other possible errors. in .NET

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where y, is the value of y calculated by the method after n steps, and other possible errors.
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Two types of instability occur in the solution of ordinary differential equations, i.e. inherent and induced instability. Inherent instability occurs when, during a numerical step-by-step solution, errors generated by any means (truncation or round-off) are magnified until the true solution is swamped. Fortunately transient stability studies are formulated in such a manner that inherent instability is not a problem. Induced instability is related to the method used in the numerical solution of the ordinary differential equation. The numerical method gives a sequence of approximations to the true solution and the stability of the method is basically a measure of the difference between the approximate and true solutions as the number of steps becomes large. Consider the ordinary differential equation
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(IV.2.3) (IV.2.4)
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with the initial conditions y ( 0 )= y o which has the solution
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y ( t )= yoel .
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Note that i, is the eigenvalue [l] of the single-variable system given by the ordinary differential equation (IV.2.3). This may be solved by a finite difference equation of the general multistep form:
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where ai and /.Ii are constants. Letting
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1 ai(z)i
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and constraining the difference scheme to be stable when A = 0, then the remaining part of (IV.2.S) is linear and the solutions are given by the roots zi (for i = 1,2,. . . ,k ) of m(z)= 0. If the roots are all different, then
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y , = A,(z,) AZ(Z2) ***A,(z,)
(IV.2.7) (IV.2.8)
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and the true solution in this case (A = 0) is given by
y(t,) = ~
~ ( 2 , )
+ o ( ~ P = yo +
where superscript p is the order of accuracy.
The principal root zl, in this case, is unity and instability occurs when lzil 2 1 (for i = 2,3,. . .,k, i # 1) and the true solution will eventually be swamped by this root as n increases. If a method satisfies the above criteria, then it is said to be stable but the degree of stability requires further consideration. Weak stability occurs where a method can be defined by the above as being stable, but because of the nature of the differential equation, the derivative part of (IV.2.5) gives one or more roots which are greater than or equal to unity. It has been shown by Dalquist [2] that a stable method which has the maximum order of accuracy is always weakly stable. The maximum order or accuracy of a method is either k + 1 or k + 2 depending on whether k is odd or even, respectively. Partial stability occurs when the step length (h) is critical to the solution and is particularly relevant when considering Runge-Kutta methods. In general, the roots zi of (IV.2.7) are dependent on the product hi, and also on equations (IV.2.6). The stability boundary is the value of hi. for which lzil = 1, and any method which has this boundary is termed conditionally stable. A method with an infinite stability boundary is known as A-stable (unconditionally stable). A linear multistep method is A-stable if all solutions of (IV.2.5) tend to zero . as n + co when the method is applied with fixed h > 0 to (IV.2.3) where j is a complex constant with Re(;.) < 0. Dalquist has demonstrated that for a multistep method to be A-stable the order of accuracy cannot exceed p = 2, and hence the maximum k is unity, that is, a singlestep method. Backward Euler and the trapezoidal method are A-stable, single-step methods. Other methods not based upon the multistep principle may be A-stable and also have high orders of accuracy. In this category are implicit Runge-Kutta methods in which p < 2r, where r is the number of stages in the method. A further definition of stability has been introduced recently [3], i.e. X-stability which is the multivariable version of A-stability. The two are equivalent when the method is linear but may not be equivalent otherwise. Backward Euler and the trapezoidal method are X-stable single-step methods. The study of scalar ordinary differential equations of the form (IV.2.3) is sufficient for the assessment of stability in coupled equations, provided that j. are the eigenvalues of the ordinary differential equations. Unfortunately, not all the equations used in transient stability analysis are of this type.