Locations of the Poles of the Stability Function

2.2 Symplecticity of Runge-Kutta Methods

2.2.5 Locations of the Poles of the Stability Function

In many cases, there exist relationships between the location of the poles of the stability function of a numerical method and the number of negative weights of the quadrature formula. In this section, we investigate these relationships for the methods constructed in Theorem 2.2.6. First, we consider some examples.

Examples

a5 Symplectic diagonally implicit Runge-Kutta methods are given by the following Butcher tableau,

b¹

2 0 0 ^%^%^% 0 b¹ ^b²² 0 ^%^%^% 0 b¹ b² ^b²³ ^%^%^% 0 ... ... ... ... ...

b¹ b² b³ ^%^%^% ^b²^s b¹ b² b³ ^%^%^% bs

and possess ^R)^z* given by

R)^z* = )1 + ^b²¹^z*)1 + ^b²²^z* )1 + ^bs²^z* )1^; ^b²¹^z*)1^;^b²²^z* )1^; ^b²^s^z*^"

as stability function. So, for symplectic diagonally implicit Runge-Kutta methods, the number of negative weights of the quadrature formula )^bⁱ^"^cⁱ*i=1" "samounts to the number of poles of^R)^z* which are included in the half-plane^")^z*^%0, and the number of positive weights equals the number of poles in the half-plane ^")^z*^&0.

b* For Gauss methods, the result is the same as for diagonally implicit Runge-Kutta methods. The stability function is given by the diagonal PadHe approximation which possesses all its poles in the half-plane ^")^z*^&0. Since all the weights ^bⁱ are positive, the situation is similar to the Jrst example.

Below, we show that for the method constructed with the help of Theorem 2.2.6 the number of positive and negative weights are determined by the location of the poles of the stability function, in the same way as in the two examples considered above.

Lemma 2.2.12

^Let^R)^z*be a rational approximation ofê^z satisfying^R)^z* =ê^z+Ô)^z^p+1* and 22.96. The rational functions M^j)^z* de;ned by

M^j;1)^z* = 1 +^z(^z^j²M^j)^z*^" )2.29*

with ⁽^j = ¹²)4^j²^;1*^;1=2, and M⁰)^z* = M)^z* is the M?obius transformation of ^R)^z* de;ned by 22.146. Then we have

# ^z ^j M^j;1)^z* = 2 and ^")^z*^%0

= # ^z ^j M^j)^z* = 2 and ^")^z*^%0

)2.30*

for ^j ^%^k = R)^p^;1*⁼2S.

Remark

We already mentioned )see )2.23** that the MVobius transformation of the stability function ^R)^z* admits a continued fraction representation. In this representation the rational function M^k)^z* is determined by the relation )2.29*. A repeated application of the preceding lemma shows that the number of solutions of the equation M^k)^z* = 2 in the half-plane^")^z*^%0 equals the number of solutions of M)^z* = 2 in the same half-plane

")^z*^% 0. And then, since the^z⁰^s satisfying M)^z* = 2 are poles of the stability function

R)^z*, we conclude that the number of poles of^R)^z* in the half-plane^R)^z*^%0 equals the number of solutions of the equation M^k)^z* = 2 in the same half-plane.

Proof. We know that we have M^k)^z* = ^zg^k^#)^z*^=f^k^#)^z*, where ^g^k^#)^z* and ^f^k^#)^z* are poly-nomials. By )2.29*, this implies that M^j)^z* = ^zg^j^#)^z*^=f^j^#)^z* with ^g^#^j)^z* and ^f^j^#)^z* still polynomials for ^j =^k^;1^"^:^:^:^"0 recursively deJned by

j;1)^z* = ^f^j^#)^z*

j;1)^z* =^f^j^#)^z* +^z²⁽^j²^g^#^j)^z*^: )2.31*

Suppose that the equation M^j)^z* = 2 possesses ⁿ solutions in the half-plane ^")^z* ^% 0.

We consider the equation

)1^;²*M^j)^z* + 4² 1

M^j;1)^z* = 2^" )2.32*

53 with ² &0^!1). By assumption, this equation possesses ⁿ solutions in the half-plane

!:^z; ^$ 0 if = 0. When the parameter increases, the solutions cannot cross the imaginary axis, because the equation can never be satisDed on the imaginary axis :see equation :2.15;;. Then the assertion is proved if we check that the solution ^z of :2.32;

which appears when goes from zero to a positive value stay in the half-plan ^!:^z; ^"

0. Inserting the equations for ^g^j;1:^z; and ^f^j;1:^z; from :2.31; into :2.32;, we get the equivalent equation

:1^; ;^g^j:^z;^z²+ 4 ^;^f^j:^z; +^z²^'^j²^g^j:^z;^!= 2^zf^j:^z;^:

Because ^f^j:0; =^g^j:0; = 1, this last equation behaves, for ^z close to the origin, like 4 = 2^z:

So a solution emerges from the origin when increases, goes in the right half-plan and can never cross the imaginary axis because of condition :2.15;.

Theorem 2.2.13

Consider a symplectic Runge-Kutta method given by the couple :^A!^b;. If the method is of order^p, then the matrix ^Apossesses at least^k = &:^p^;1;⁼2)eigenvalues in the half-plane ^!:^z;^.0.

Proof. If the method is of order^p, we know that there exists a ^W-transformation, such that the matrix ^Y has the form :2.10;, with ^k = &:^p^;1;⁼2). In proof of Lemma 2.2.12 we saw that there exists at least ^k solutions of the equation O:^z; = 2 in the half-plan

!:^z;^.0. Each such solution is a pole of the stability function ^R:^z; and, since

R:^z; = ^det:^I^;^zA+^z1l^b^T;

det:^I^;^zA;

:see Hairer & Wanner &23) IV.3 proposition 3.2;, the assertions follows.

In the next lemma, we investigate the relationships between the location of the poles of the stability function ^R:^z; and the number of negative weights ^bⁱ of the quadrature formula in the special case where the method is constructed as shown in Theorem 2.2.6.

Theorem 2.2.14

For the symplectic Runge-Kutta methods constructed in Theorem 2.2.6, the number of poles of the stability function lying in the negative half-plane ^!:^z;^$0equals the number of negative weights ^bⁱ in the quadrature formula.

Proof. By Lemma 2.2.12, we already know that the number of poles of ^R:^z; in the negative half-plane equals the number of roots of the polynomial⁶ :^z; =^zg :^z;^;2^f :^z;, where ^f :^z; and ^g :^z; are deDned by :2.24;. Since ⁶ :0; ⁶= 0, and the negative half-plane is invariant under the transformation ^z ^!1^=z, the polynomial⁶:^z; = ^g:^z;^;2^f:^z; possesses the same number of roots in the negative half-plane, where ^f:^z; and ^g:^z; are deDned similarlyas in :2.25;. Noticethat this polynomial does not vanish on the imaginary axis.

The key of the proof lies in the fact that applying the transformation ^z = ^{w +1}^{w ;1} and Theorem 3.3 of 813: show that the number of roots of^!>^z? in the negative half-plane, equals the index of negativity of the bilinear form determined by the matrix ^C. The index of negativity is independant of the choice of the basis in which the bilinear form is expressed

>Sylvester's law, see Lang 832:? and equals the numberof^;1⁰^son the diagonal of I because of the relation >2.19?, and hence the number of negative ^b⁰ⁱ^s because^W^T^B^W = I.

If we consider Runge-Kutta-methods without any additional assumptions, we can see that such a theorem cannot hold. The main problem appears when the stability function of the Runge-Kutta method is not irreducible,i.e. when the Runge-Kutta method has ^s stages and the polynomial ^P>^z? in >2.9? is of order less than ^s. Since in this case the relationship between the signs of the weights ^bⁱ and the scalar product deNned by the matrix ^C in >2.16? >see the proof of the theorem below? is no longer evident.

Moreover, only irreducible Runge-Kutta methods must be considered, otherwise we can construct Runge-Kutta methods with parasitic weights ^bⁱ which do not inQuence the stability function.

Theorem 2.2.7

Consider a symplectic s-stage Runge-Kutta method where the stability function given by 92.9= is irreducible with ^P>^z?a polynomial of order^s. Then, the number of negative weights ^bⁱ of the quadrature formula amounts to the number of poles of the stability function in the negative half-plane ^!>^z?⁽0.

Proof. By assumption, the polynomials^P>^z? and^P>^;z? have no roots in common. Then the polynomials ^f>^z? and ^g>^z?, deNned by

g>^z?

f>^z? = T>1

?⁺

where T>^z? is the MUobius transformation of the stability function ^R>^z?, are polynomials of degree^s and ^s^;1 respectively, and have no roots in common. Following Theorem 3.2 of 813:, this implies that the matrix ^C= >^c^ij?i$j=1$:::$s deNned by >2.16? is invertible. If we proceed as in Lemma 2.2.9, we see that the matrix^C is given by

C= >^W^L?^T^B>^W^L?⁺

where >^W^L? is invertible. Positive and negative ^b⁰ⁱ^s are then related to the linear spaces on which the scalar product deNned by the matrix^C is positive or negative >Sylvester's law?. To conclude, we proceed as in Theorem 2.2.14 to see that the number of poles of the stability function ^R>^z? in the negative half-plane ^!>^z?⁽0 equals the index of negativity of the bilinear form determined by the matrix ^C.

It remains to treat the case where the stability function given by >2.9? is the quotient of two polynomials of degree ^r less than the number of stages ^s of the method. If we consider diagonal implicit Runge-Kutta methods, we notice that the number of negative weights ^bⁱ equals the number of poles in the negative half-plane ^!>^z?⁽0 plus >^s^;^r?⁼2.

Roughly, if^det>^I^;zA? and^det>^I+^zA? have roots in common, then half of them >>^s;r?⁼2?

belong in the half-plane^!>^z?⁽0 and the other in the half-plane^!>^z?⁶0. If we consider 3-stage Runge-Kutta methods of order at least 2 which admit the Nrst diagonal PadYe approximation >^1+'=2^1;'=2? as stability function, we have only one negative weight ^bⁱ and the other two positive. Since the Nrst diagonal PadYe approximation possesses a pole in the

55 half-plane )^z*^!0, we observe the same behavior as for diagonal implicit Runge-Kutta methods. It seems reasonable to conjecture that for symplectic Runge-Kutta methods the pole locations of the stability function determine the number of negative and positive weights, in the same way as in the examples discussed above.

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