The exact region of stability for MacCormack scheme. Hoon Hong. Research Institute for Symbolic Computation. Johannes Kepler University

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1 The eact region of stailit for MacCormack scheme Hoon Hong Research Institute for Smolic Computation Johannes Kepler Universit A- Linz, Austria Astract Let the two dimensional scalar avection equation @u = ^a + ^ : We prove that the stailit region @ MacCormack scheme for this equation is eactl given ^a t t + ^ 1 where t ; and are the grid distances. It is interesting to note that the stailit region is identical to the one for La-Wendro scheme proved Turkel. 1 Introduction 1 a + a + = 1=8 a = + = = 1 Stailit analsis is one of the most fundamental tasks in devising reliale numerical methods for solving partial dierential equations. In this paper, we investigate the stailit of MacCormack's scheme [8] which is widel used in solving hperolic partial dierential equations, especiall in aerodnamics. In particular we consider the following scalar two dimensional = : In [1] Wendro derived the following sucient condition for the stailit region: a + a + 1=8: where a and are respectivel ^a t and ^ t, where again t ; and are the grid distances. As it was remarked [1], numerical sampling shows that it is not a necessar condition. The main result of this paper is the derivation of the following necessar and sucient condition: a + Figure 1 compares the region otained Wendro and the one otained here. It is interesting to note that the eact stailit region of MacCormack scheme derived here is identical to that for La- Wendro scheme proved Turkel [11]. In fact, one will notice in the susequent sections that the underling proof techniques are similar. This research was done in the framework of the European project ACCLAIM Figure 1: Stailit Regions The structure of the paper is as follows. In the net section, we will give a precise description of the stailit prolem and the eact solution for MacCormack scheme. In the following section, we will give a proof. Stailit of MacCormack Scheme In this section, we recall the denitions of the La-Wendro scheme and the MacCormack scheme. Then we give a precise statement of the stailit prolem for the MacCormack scheme and the solution for it. Let T and T e the translation operators respectivel de- ned T u(; ; t) = u( + ; ; t); T u(; ; t) = u(; + ; t): Let L = I + 1 a(t? T?1 ) + 1 (T? T?1 ) + 1 a (T? I + T?1 ) + 1 (T? I + T?1 ) + 1 a(t? T?1 )(T? T?1 ): 1

2 Then the La-Wendro [6] scheme is given Let u(; ; t + t ) Lu(; ; t): G ++ = I + a(t? I) + (T? I); G?? = I + a(i? T?1 ) + (I? T?1 ); G?+ = I + a(i? T?1 ) + (T? I); G +? = I + a(t? I) + (I? T?1); M 1 = 1 (I + G ++G?? ); M = 1 (I + G?+G +? ); M = M M 1 : Then the MacCormack scheme [8] is given u(; ; t + t ) Mu(; ; t): As oserved in [1], the two operators M and L are related M = L? Q where Q = 1 a(t? I + T?1 )(T? I + T?1 ): In order to otain the amplications for the La-Wendro scheme and the MacCormack scheme, one replaces T and T with e i and e i respectivel, otaining r = 1? a (1? cos )? (1? cos )? a sin sin s = a sin + sin q = a(1? cos )(1? cos ) l = r + is m = l? q where l and m are respectivel the amplications of the La- Wendro and the MacCormack scheme. Now the stailit prolem for the MacCormack scheme is to nd a necessar and sucient condition on a and such that for all real values of and the asolute value of m is not greater than 1. More formall, the prolem is to nd a quantier-free formula F (a; ) which is equivalent to the quantied formula 8(; ) R jmj 1: Let us oserve that we can remove the trigonometric functions from the aove formula carring out the usual parameterization with u 1 = sin, v 1 = cos, u = sin and v = cos, along with the condition u 1 + v 1 = u + v = 1. The resulting formula (involving onl algeraic operations) elongs to the language of the rst order theor of real closed elds. It is well known that the quantier elimination in that theor can e carried out algorithmicall [1]. Further, there have een various reakthroughs in devising more ecient algorithms [1,,, 5,, 9]. There also eists an implementation []. Thus, in principle, the stailit prolems can e solved completel automaticall a computer program. In fact, there have een interesting and important progress in this direction as reported in [7] where Liska and Steinerg succeeded in deriving the stailit conditions for several prolems using the quantier elimination program qepcad []. But, in spite of these various progresses, the complete automation for the MacCormack scheme is not et feasile due to the huge requirement of computing times. Thus, the result reported in this paper is ased on \human-rain" proof. But it must e emphasized that during conjecturing the main result, computer plaed essential role, as will e reported in some other paper. Now we state the main result of this paper, that we will prove in the net section. Theorem 1 (Stailit of MacCormack Scheme) 8(; ) R jmj 1 a + Proof In this section, we give a proof of the theorem stated aove. The proof will consists of proofs of four lemmas. The dependenc among the lemmas is as follows. The rst lemma is used for proving the second lemma. The second lemma is used for proving the third and the fourth lemma. The third lemma proves that the proposed stailit condition is necessar. The fourth lemma proves that it is also sucient. Slightl dierent variations of the rst two lemmas were proved Turkel [11]. Thus, the reader who is familiar with [11] might want to skip reading the proofs of the two lemmas. We decided to include the complete proof here ecause our lemmas are slight dierent from Turkel's in that ours do not have asolute value function in the formulation, and thus require careful adjustments in the proofs. Lemma 1 (Univariate quantier elimination) Let P () = (a + )? a? : 8 R P () a + Proof. The proof will e case analsis. First three cases take care of the \degenerate" situations. Case: a = L.H.S. 8 R? Case: =? (? 1) L.H.S. 8 R a? a 8 R a (a? 1)

3 Case: a = 1 a (a? 1) a 1 a + 1 L.H.S. 8 R + 6 +? = (1) + 1 Case: Otherwise, that is, a 6= ; a 6= 1; 6=. Note that P () = (a? a ) + : Since (a? a ) is not zero, P is of degree in. Thus we have where L.H.S C1 ^ C C1 : a? a C : 8 R [ P () = =) P () ]: We see immediatel C1 a 1 a < 1 ecause a 6= 1. Let us now eliminate the quantier from C. C 8 R [ a(a + )? a = =) P () ] 8 R [ (a + )? a = =) P () ] 8 R [ a +? a 1 = =) P () ] 8 R [ = =) P () ] P : Finall we otained a quantier-free formula. Let us simplif this formula. P a +? a?! a 1? a? a 1? a? a? a? a? a? 1? a? a? a a? 1? a? 1? a a +? 1 : In order to link the second and the third steps from the ottom, we have used the elementar fact that (u? v ) has the same sign as (u? v). Now putting together, we nall otain L.H.S. C1 ^ C a < 1 ^?1? a a < 1 ^ a +? 1 a +? 1 a +? 1 Remark: This univariate quantier elimination prolem is small enough to e carried out a computer program. In fact, we have tried the quantier elimination program qepcad []. Given the input 8 (a + )? a? ; the program, in aout 5 seconds on a DEC station, outputs the quantier-free formula: a 6 + a + a + 6? a + 1a? + a + 1: B suitale rewriting, one see that it is equivalent to a + 1: Remark: It is interesting to note that the polnomial shown aove a 6 + a + a + 6? a + 1a? + a +? 1 is the discriminant of the input polnomial P () = (a + )? a? with respect to upto some trivial factor. Thus, for this particular case, the discriminant computation was sucient. It is ecause P has onl one real root, sa. In that case, it is well known that P is the discriminant of P () upto some constant multiple. But it is not true in general when P has more than one real root. Lemma (Bivariate quantier elimination) Let Proof. P (; ) = (a + )? a? : 8(; ) R P (; ) a + L.H.S. [ 8 R P (; ) ] ^ [ 8 R8 R? fg P (; ) ] [ 8 R (a? a ) ] ^ [ 8 R8 R? fg a +? a?! ]

4 a 1 ^ 8z R (az + )? a z? a 1 ^ a + 1 ( Lemma 1) a + 1 Lemma (Necessar condition) 8(; ) R jmj 1 =) a + Proof. The proof will e case analsis. The rst three cases will take care of \degenerate" situations. Case: a = and = We see that r = 1; s = ; q =. Thus, jmj = 1. Hence we have L.H.S. 8(; ) R R.H.S Case: a = and = 1 We see that r = cos ; s = sin ; q =. Thus, jmj = 1. Hence we have L.H.S. 8(; ) R (1) 1 R.H.S Case: = and a = 1 We see that r = cos ; s = sin ; q =. Thus, jmj = 1. Hence we have L.H.S. 8(; ) R 1 1 (1) + 1 R.H.S Case: Otherwise B epanding jmj with respect to and around (; ), we otain Let jmj = 1 + 1? (a + )? a? + O( ): P (; ) = (a + )? a? : One can easil verif that P is a non-zero polnomial (i.e. does not vanish identicall). 1 Thus, the ehaviour of jmj near the origin is determined that of P. Hence we have L.H.S. =) 8(; ) R P (; ) a + 1 ( Lemma ) R.H.S 1 In fact, the rst three cases were introduced in order to lter out the situation when P is a zero polnomial. Lemma (Sucient condition) Proof. 8(; ) R jmj 1 (= a + Let us rst note jmj = jl? q j Thus we have = (r + is)? q j = jr? s? q + rsij = (r? s? q ) + r s = r + s + q? r s? r q + s q + r s = r + s + q + r s + r q + s q? r q = (r + s + q )? r q : 8(; ) R jmj 1 8(; ) R (r + s + q )? r q 1 (= 8(; ) R (r + s + q ) 1 8(; ) R r + s + q 1: (1) A digression to tell an inside stor. Aove, I took a \old" step of ignoring the term r q. It is hard to elieve that the proof will go through without the term, ut it turns out to e so. In fact, I have spent several months, in vain, searching for proofs which take account of the term ecause I was preconditioned to the idea that the term must pla an essential role. Onl after drawing lots of pictures using plotting programs and inspecting lots of computation traces using computer algera sstems, one morning, I \accidentl" hit upon the intuition that the term might not pla an important role for stailit. Then I could reach, after onl few more hours of work, the end of the proof, to m jo! The end of the digression. Since and alwas occur as arguments of trigonometric functions, we can restrict their ranges. 8(; ) R r + s + q 1 8(; ) [?; ] r + s + q 1: Here the choice of [?; ] instead of [; ] is intentional as will e seen now. Let?1 ; 1 cos ; cos 1 = sin ; = sin : cos = p 1?, cos = p 1?. (If we had chosen [; ] as the range of and instead, we could have had dicult since we could not have determined the sign of cos and cos from and.) We rewrite r; s; and q in terms of and, otaining r = 1? a?? a p 1? p 1? s = ap 1? + p 1? q = a :

5 After some straightforward calculation, we otain w 1 (; ) = r + s + q Now oserve = 1? a? +a + +8a +16a (1?? + ) +16a p 1? p 1? +16a p 1? p 1? : 8(; ) [?; ] r + s + q 1 8(; ) [?1; 1] w 1 (; ) 1: () 1?? + = 1? (1? )? (1? ) 1: Thus we have 8(; ) [?1; 1] w 1 (; ) 1 (= 8(; ) [?1; 1] w (; ) 1 where w (; ) is otained from w 1 (; ) replacing the term (1?? + ) with 1, that is, w (; ) = 1? a? +a + +a +16a p 1? p 1? +16a p 1? p 1? : Let w (; ) e otained from w (; ) dropping the term p 1? p 1?, that is, w (; ) = 1? a? Now we claim that +a + +a +16a +16a : 8(; ) [?1; 1] w (; ) 1 (= 8(; ) [?1; 1] w (; ) 1 (5) In order to prove this claim, let us assume that 8(; ) [?1; 1] w (; ) 1: Let ( ; ) [?1; 1] e aritrar ut ed. show that w ( ; ) 1: We will do so case analsis. Case: a Clearl, we have w ( ; ) w ( ; ): we need to From the hpothesis, we also have Thus, we have Case: a < Clearl, we have w ( ; ) 1: w ( ; ) 1: w ( ; ) w ( ;? ) w ( ;? ): From the hpothesis, we also have Thus, we have w ( ;? ) 1: w ( ; ) 1: This completes the proof of the claim that we can drop the radical terms. Now note that 8(; ) [?1; 1] w (; ) 1 8(; ) [?1; 1]?a? + (a + ) (= 8(; ) R? a? + (a + ) a + 1 ( Lemma ): (6) Finall, appling the transitivit of the logical implication ((=) to the implication relations in (1),, (),, (5) and (6), we conclude that 8(; ) R jmj 1 (= a + From Lemmas and, Theorem 1 immediatel follows. Concluding Remarks In this paper, we have proved that the stailit region of the MacCormack scheme for the two dimensional scalar avection equation is eactl given a + As Turkel [11] has done for the La-Wendro scheme, it seems to e possile to generalize the result of this paper to n- valued functions where a and are commuting real smmetric matrices. This will e investigated in a susequent note. Though it is not et computationall feasile to automate the whole process a computer program, some part of the proof could e alread carried out computer programs. Furthermore various conjecturing processes were greatl aided computer programs. These eperience will also e reported in another paper. Acknowledgment: I would like to thank Stanle Steinerg and Richard Liska ecause the introduced me to the stailit prolem of MacCormack scheme and also rought Wendro's work to m attention. 5

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