From: <ÓÉ Windows Internet Explorer 8 ±£´æ> Subject: Accuracy and Stability Date: Thu, 24 May 2012 11:19:33 +0800 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01CD399F.181B0660" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.6157 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01CD399F.181B0660 Content-Type: text/html; charset="gb2312" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.cs.sjsu.edu/~rucker/capow/paper/node3.html Accuracy and Stability


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Accuracy and = Stability

Because of the discrete nature of computers, we can only approximate = the=20 solution to a PDE at certain grid points on the 3D"" plane. We = label=20 these points of approximation 3D"" where 3D""=20 and=20 3D"" and employ = the=20 abbreviations 3D"" and 3D""=20, where=20 3D"" is the = numerical=20 approximation at the grid points of interest for 3D"", the = actual PDE=20 solution. We are interested, of course, in making 3D"" as small = as=20 possible. To accomplish this we must consider the accuracy and = the=20 stability of our approximation scheme.=20

The accuracy of a scheme is really just a matter of minimizing the = error term=20 in the Taylor appoximation for the scheme. Because of its simplicity --- = and its=20 applicablity to eqns (7)-(9) --- we will use as an example the first = order=20 linear traveling wave equation=20

3D""

From Taylor's formula we have that=20

3D""

where 3D"" and also = that=20

3D""

where 3D"", so it = follows that=20

3D""

since the 3D"" and the = 3D""=20 terms=20 cancel by applying eq (10). This suggests the following numerical = algorithm to=20 simulate eq (10):=20

3D""

where 3D"". This = algorithm=20 explicitly shows how to compute values for a new time, 3D"", given = information=20 at the current time, 3D"". To = compute the=20 value at a fixed time 3D"", the = algorithm must=20 be run through 3D"" time = steps. Since=20 the error from each step is 3D"", the total = error=20 between the computed solution and the actual solution at 3D""=20 is =20 --- assuming that the errors from each time step can be added together = (which is=20 not the case when the algorithm is unstable as we will soon discuss). = Due to the=20 form of the total error (if the algorithm is stable), we refer to the = algorithm=20 as being ``first order accurate in x and first order accurate in=20 t". If, instead of using the Taylor expansion in eq (11), we use=20

3D""

we end up with a similar numerical scheme that also is first order = accurate=20 in both x and t:=20

3D""

The scheme in eq (14) is called a downwind scheme; the = scheme in eq=20 (15) is called an upwind scheme. We could also subtract eq (15) = from eq=20 (11), which yields=20

3D""

which implies that the numerical scheme=20

3D""

is first order accurate in time and second order accurate in space = since the=20 error at a fixed time value is 3D"". = Obviously, one=20 wishes to employ schemes with higher order errors since these schemes = require=20 less grid points (and therefore less computational time) to obtain a = computed=20 solution that is within a given error of the actual solution.=20

There is, however, more to numerical simulation of PDEs than just = Taylor=20 expansions and accuracy questions. Consider eq (10) where the domain of = x=20 is infinite and the initial condition 3D"" is = specified:=20

3D""

3D""

The unique solution to this equation is simply=20

3D""

In other words the value of the solution is constant along each line = x=3Dat=20 + C. These lines are called characteristic lines, and they occur = when the=20 domain of x is finite as well as infinite. In other words the = value of=20 the solution at a point depends only on which characteristic line = crosses=20 through that point. If we consider x to be the abscissa and = t the=20 ordinate, then each characteristic line has a positive slope of 3D""=20. The=20 constant a is often referred to as the wave speed since = it=20 specifies how fast a disturbance at a point x propagates down to = other=20 values of x. Now consider what happens when we apply the downwind = scheme=20 in eq (14) in an attempt to solve eq (10). The downwind scheme only uses = the=20 points 3D"" and 3D""=20 to=20 estimate the value of the point 3D"", but we = know from=20 the characteristic lines that the value at this point only depends on = the value=20 of the solution at 3D"" if 3D""=20, which=20 is outside the range of the two points being used. This causes the = computer=20 generated solution using the downwind scheme to go haywire and yield = gibberish.=20 One may wonder how can this happen since we know the scheme is accurate. = The=20 answer is that accuracy only considers local error, whereas the = stability problem that is occuring in this case is a global issue.=20

The Taylor series analysis of error in determining accuracy = assumes=20 that the solution is correct at 3D"", and then = finds the=20 error that is created when the solution at 3D"" is = approximated.=20 However, if the error accrued in a step is magnified by each subsequent = step, we=20 can no longer just add the individual errors produced at each step; in = fact,=20 when the errors are magnified by subsequent steps, we observe that the = solution=20 is unstable, that is, the global error quickly blows up. When a scheme = attempts=20 to approximate the solution at a point\ 3D"" whose = characteristic=20 line crosses the 3D"" line = outside the=20 range of points used by the scheme, then the scheme, no matter how = accurate,=20 will always magnify errors in each step and therefore be unstable. In = other=20 words the numerical range of dependence must always contain the = theoretical=20 range of dependence of the solution or instablity will occur. This = principle is=20 called the CFL (Courant-Friedrichs-Lewy) condition (sometimes referred = to simply=20 as the Courant condition) [7]. This guarantees that the downwind scheme = will=20 always be unstable and that the upwind scheme can only be stable if 3D""=20. Even=20 if a scheme satisfies the CFL condition, it may still be unstable, but = we will=20 not be analyzing any schemes that possess this flaw in this paper.=20

Before analyzing the FPU nonlinear wave equations, we wish to apply = the above=20 analysis to the linear wave equation, 3D"", where we = use =20 to denote the constant 3D"" for = reasons that=20 will soon be apparent. If we define 3D"" where 3D""=20 and=20 3D"" then we = have that=20 the wave equation is satisfied, 3D"", and also = that the=20 mixed partials are equal, 3D"". This = allows us to=20 write the wave equation as the following first order partial = differential=20 system:=20

3D""

Eq (21), of course, is just the matrix form of the travelling wave = equation=20 from eq (10), and its solution contains many of the same properties as = eq (10).=20 As opposed to the solution depending upon one characteristic line, the = solution,=20 3D"", now = depends on two=20 characteristic lines whose wave speeds are c and -c, the=20 eigenvalues of 3D"". = Therefore the=20 slopes of the two characteristic lines are the reciprocals of the wave = speeds,=20 3D"" and 3D""=20. As=20 before, we must satisfy the CFL stability condition, so we now require = that .=20

All that remains is to find a sufficiently accurate scheme. To obtain = an=20 approximation for 3D"" we add eq = (11) to=20 eq (15), yielding=20

3D""

which we can combine with the analogous expression from the Taylor = expansions=20 with respect to time to yield the numerical scheme=20

3D""

which we can reexpress as=20

3D""

We note that this scheme is 2nd order accurate with respect to both = time and=20 space.=20

We are now ready to consider the nonlinear FPU equations. We begin = with the=20 quadratic FPU equation=20

3D""

where 3D"" and 3D""=20. (Note=20 that 3D"" since = 3D""=20, like=20 the spring constant k, is always positive in any realistic spring = model.)=20 As with the linear wave equation we can rewrite eq (24) in vector form, = but now=20 the matrix A will depend upon the value of 3D"":=20

3D""

The wave speeds of the two characteristics are still the eigenvalues = of=20 A, which are 3D"". (Note = that we=20 require 3D"" to be = positive=20 since otherwise the wave speeds would be imaginery and the partial = differential=20 equation would no longer be hyperbolic, which leads to equations beyond = our=20 analysis here and causes the scheme to become unstable). Since the = speeds are=20 functions of 3D"", that = means the=20 speeds vary, which implies that the characteristics are now curves whose = slopes=20 vary as they progress in time as opposed to being straight lines as was = the case=20 in the linear wave equation. This leads to a stability problem as the = CFL=20 condition now requires that 3D"". If 3D""=20 is=20 small compared to 3D"", then = 3D""=20 will=20 be positive and the variations in 3D"" may not = be severe=20 enough to violate the CFL condition, but if the nonlinearity becomes = more=20 significant, which means the value of 3D"" becomes = larger,=20 eventually the CFL condition will be violated (or 3D"" will = become=20 nonpositive) and the scheme will become unstable. (Note: in the = experiments in=20 section 5, we will set 3D"" and use = values of=20 3D"" that are = slightly=20 smaller than 1 which will cause the CFL condition to be violated before = =20 becomes nonpositive.) Since by definition 3D"", we see = that if=20 3D"" gets = large not only=20 does the numerical scheme become unstable, but also the underlying = differential=20 equation becomes less valid since higher order terms in the Taylor = approximation=20 given in eq (4) start to become significant.=20

The analysis for the cubic FPU equations is similar. We now have that =

3D""

where 3D"", which = can be=20 expressed in matrix form as=20

3D""

The corresponding CFL condition is 3D"". Note = that if =20 is positive, the wave speeds will be real and we will always have a = hyperbolic=20 differential equation.=20

There are numerous accurate schemes that one can apply in the region = where=20 the algorithm is stable. To simulate the quadratic FPU equation, eq (8), = we can=20 combine eq (17) with eq (23) to form=20

3D""

which can be rewritten as a scheme which is second order accurate in = time and=20 space:=20

3D""

Eq (28) is a generalization of the FPU quadratic scheme (eq = (ii)).=20 After setting 3D"" and 3D""=20, one=20 can quickly establish that the right hand sides of eq (28) and eq = (ii)=20 are identical.=20

Similary, we can simulate the cubic FPU equation, eq (9), by again = combining=20 eq (17) with eq (23) to obtain the following second order accurate = scheme:=20

3D""

The fact that the scheme in eq (30) is not the same as FPU's cubic = scheme (eq=20 (iii)) does not pose a problem. Either FPU's scheme or eq (30) = can be=20 used to simulate the cubic FPU equation as both schemes are second order = accurate and will therefore produce essentially equally valid = simulations. (The=20 second order accuracy of the FPU's cubic scheme is quickly established = using=20 Taylor series.)=20

There is another phenomenon that is important in nonlinear hyperbolic = partial=20 differential equations. Because the characteristics are curves, = characteristics=20 within a family can intersect each other. (In the linear case, each = family of=20 characteristic lines has a constant slope: 3D"" for one = family,=20 3D"" for the = other=20 family.) When the characteristics intersect, a shock, i.e., a=20 discontinuity in the solution, occurs. While multiple weak solutions to = the=20 nonlinear equation are possible, one usually desires to simulate the = unique=20 observed solution that the phenomena being modeled exhibits. For most = physical=20 phenomena the solution of interest is the Lax entropy solution = of the=20 equation, which can be simulated using monotone schemes [8]. A = particularly nice=20 introduction to shock capturing monotone schemes is presented in [9].=20

We do not pursue these schemes here, however, as we are interested in = other=20 aspects of the transition to instability.=20


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Rudy Rucker
Sat May 9 20:21:15 MET DST 1998=20
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