Mixed methods for time integration

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 17118 (1979) 259-275 0 NORTH-HOLLAND PUBLISHING COMPANY

MIXED METHODS FOR TIME INTEGRATION

T. BELYTSCHKO, H.-J. YEN and R. MULLEN Department of Civil Engineering, Northwestern University, Evanston, Illinois 60201, USA

The time integration of fluid-structure and soil-structure problem is often quite uneconomical when performed with a single integration method and a single time step. Three techniques for enhancing computational efficiency are presented: explicit-implicit partitions, explicit-explicit partitions with different time steps, and implicit-implicit partitions. The latter two require interpolation and extrapolation on the interface. A model problem is presented for examining the stability of these procedures and stabilization techniques are examined.

1. Introduction

As a consequence of safety issues, a considerable need has arisen for the transient analysis of fluid-structure and soil-structure systems. The numerical analysis of these problems is usually accomplished by semidiscretization methods: the problem is discretized in space by a finite element or finite difference method, and the resulting ordinary differential equations are then integrated in time. There are basically two classes of algorithms for the time integration of the semidiscreti- zations: explicit and implicit. Explicit methods usually do not involve the solution of any equations, so fewer computations are needed per time step, but numerical stability requires that the time step be small. Implicit methods, on the other hand, involve the solution of a system of equations, but much larger time steps can be employed because they are unconditionally stable for linear and many nonlinear problems, these issues are elaborated on in [ 1 I . The solution of the governing equations in implicit methods can be accomplished by either direct solution or iterative methods; the latter require less core storage, particularly for large meshes, but for structural meshes their convergence rates tend to be unacceptably poor.

Although any engineering problem can in principle be integrated entirely with a single method and a single time step, such computations are in fact often quite uneconomical for media-structures problems. Implicit time integration with a direct solution procedure requires too much core and too many computations because of the large size of media meshes, particularly in three dimensional problems. However, the convergence of iterative implicit methods is very slow for structural meshes. Thus, an effective implicit integration would involve the use of an iterative procedure in the media mesh, a direct solution procedure in the structure mesh. Since it would be extremely expensive to obtain a direct solution in the structure for each iteration of the media mesh, the two partitions would have to be integrated separately in a staggered manner. This procedure will be called an implicit-implicit partition (I-I).

Straight explicit time integration of a structure-media mesh is also inefficient, for the stability limit arising from the structural mesh is far smaller than that arising from the media mesh.

260 T. Belytschko et al./Mixed methods for time integratiorz

Belytschko and Mullen [ 21, [ 31 have recently presented an explicit-implicit partition where part

of the mesh is integrated explicitly, e.g. the media mesh, and part is integrated implicitly, e.g. the structure. They have shown that the stability limit on the time step is determined strictly by the

highest frequency in the explicit partition. An alternative approach presented by Belytschko and Mullen [4] is to integrate both partitions explicitly, the flexible part with a time step At. the stiff

part of the mesh with a smaller time step At/m. This procedure, which is called an E”’ E parti-

tion, enhances the efficiency of explicit time integration dramatically for fluid-structure prob-

lems, for At is then determined by the highest frequency in the media mesh. A significant feature of I - I and E” - E partitions is that the difference equations do not

suffice to establish all of the required variables in the time integration procedure. As is shown sub-

sequently, in an I - 1 partition, an extrapolation is required on the nodes on the interface between the partitions, while in an E” -- E partition, interpolation is required along the interface. These

extrapolations and interpolations effect the stability of the time integration. The study of these mixed partitions in time integration has been quite limited. Belytschko and

Mullen [3] have proven the conditional stability of explicit-implicit partitions using energy methods and showed that the time step is limited strictly by the maximum frequency in the explicit

partition of the mesh. Hughes and Liu [ 5 1 have proven a similar stability condition for their alter- nate explicit-implicit approach. Park et al. [ 61 have studied implicit-implicit partitions where a structure is integrated implicitly in conjunction with a system of dampers that represents a doubly asymptotic approximation for an infinite body of fluid. Through the use of a model problem. they studied the stability of various extrapolations and proposed stabilization techniques.

In this paper, mesh partitions will be reviewed and a model problem is presented for the purpose of elucidating the stability properties. It is shown that the stability properties of explicit- explicit partitions depends on the choice of the interpolation: for a linear interpolation, the method is conditionally stable. Implicit-implicit partitions are shown to be unconditionally unstable in the absence of damping. However, it is shown that through the introduction of damping

in the interface elements, unconditional stability can be achieved in the model problem. Some re-

sults which illustrate the applicability of these findings to more elaborate meshes and nonlinear

problems are then presented.

2. Governing equations

The equations of motion which result from a semidiscretization of the momentum equations

can be written in the form [71, [81

Mii=f,

where

f=f”“‘-fin1

fi”j = j-B% d V. V

(2)

(3)

T. Belytschko et &/Mixed methods for time integration 261

Here a superscript dot denotes a time derivative, t the transpose of a matrix and M = mass matrix, u = column matrix of nodal displacements, f”“’ = p = external nodal forces, fint = internal nodal forces, B = strain-displacement matrix, e = stresses.

For a linear, undamped system

f int=Ku (4)

so that eq. (1) can be written as

A& +Kk!# =f-, (5)

which is the standard governing equation for linear structural dynamics. For the purpose of considering a partitioned time integration, we now subdivide the mesh into

subdomains A and B, each of which is to be integrated by a different method. The equations of motion can then be partitioned in the form

Similarly, the linear equations can be partitioned as

(6)

(7)

A subdivision such as indicated by eqs. (7) is depicted in fig. 1. It can be seen that when the nodes are partitioned into two groups, A and B, then the elements fall automatically into three groups

R, : elements with all nodes in A,

% : elements with all nodes in B, R AB: elements with nodes in both A and B.

Neither A nor B need be physically contiguous groups of nodes, but in structure-media problems it is usually natural to partition the mesh into the medium and structure.

We will restrict our attention to time integration by linear multistep methods. Implicit linear multistep formulas will be written in the form

u n+l = poll”+’ + h”, 03)

262

Fig. 1. Partition of mesh.

where the superscript denotes the time step, PO is a scalar factor which depends on the type of multistep formula and the time step, and h” denotes the historical values of u and its time deriv- atives, i.e. those values pertaining to time step n or the preceding time steps.

Explicit linear multistep equations are written in the form

g+l = Qj” + (Y,u” + h”-“. (9)

If part of the mesh is to be integrated with a smaller time step, At/m, then the explicit formula is

Un+Ipl = PP - “n+IP-1 I + + n+[P-f t + hn+[p--a] 7 (10)

where the brackets denote a fractional time increment, i.e.

g+lPl = p+pim - - u(nAt + p/w2 At).

3. Equations for partitioned systems

We will now consider the combinations of time integration shown in table 1.

Table 1. Integration partitions

Designation Time integration in Time integration in partition A partition B

E-E explicit with A8 explicit with At Em - E explicit with At/m explicit with Aht I-I implicit with Ai implicit with At E-I explicit with At implicit with At

__~~------ ___~-..----- _-____--- -

This list is not exhaustive, and numerous other possibilities of practical impo~ance exist.

3.1. E - E Partition

(11)

The E - E partitions are primarily of value when two different programs are used for different

T. Belytschko et aLlMixed methods for time integration 263

parts of the mesh. For example, a common application is to use an explicit Lagrangian finite difference program such as HEMP 193 or REXCO [lo] with an explicit structural finite element program such as BUS [ 111. The equations for updating the displacements are obtained by substituting eq. (9) into (6), which yields

If a consistent mass matrix is used, uz+l and u:+’ are coupled through MAB, and a completely independent solution of the two mesh partitions is impossible. If the mass matrix is lumped, and hence diagonal, MAB vanishes, so

UA n+l = /31AM-pf; + cYIAU~ + hy (134

From the above, it can be seen that the right hand side depends only on the results of time step n and preceding time steps, so either partition may be integrated first, independent of the other partition. Furthermore, eqs. (13) corresponds to a precise application of the explicit time integration formulas to each subdomain, so stability is governed by the usual criteria. For example, if the central difference equations are used for a linear system, then

2 At< -

w ’ max (14)

where w,, is the maximum eigenvalue of

Ku = w2Mu, (19

3.2. E - I Partition

Let A be the explicit partition, B the implicit partition. Consider the equations of motion at time step n in A, time step n + 1 in B. With a lumped mass matrix, substitution of eqs. (8) and (9) into eq. (6) then yields

(16b)

The right hand side of eq. (16a) depends only on historical values, so u:+r can be determined independent of ug”. Equation ( 16b) represents a nonlinear set of algebraic equations for a:”

264 T. Belytschko et aLlMixed methods for time integration

since fi+’ is a function of u;f+ ’ ; moreover, fg+’ depends also on the stresses on the interface elements RAB, so it is also a function of 24z+l. Hence eq. (16b) can not be solved until eq. (16a) is completed, which means that the explicit subdomain must be integrated first.

This interrelationship is more obvious if we consider the linear counterpart of eqs. ( 16)

u;+’ = fllAM~‘(p; - K,u; -- K,,u;) + q/p; + hi-’

n+1 = ul3 &M, +K, p;+l - K;,u;+~

OB

( 17a)

(17b)

As can be seen from the above, all of the terms on the right hand side will be known if the time

integration is performed in the order given; explicit first, followed by implicit. It has been proven

in [ 31 that this procedure is conditionally stable: the time step must be bounded by eq. (17) with

w mal the maximum frequency of the explicit partition with all nodes associated with R, and R,, unconstrained.

3.3. E” - E Partition

Let partition A be integrated with a time step At/m, partition B with a time step At. Substitu-

tion of eqs. (9) and (10) into eq. (6) gives

u;+l = PIBM;‘f;; + qBu;; + hi-’ (18a)

.;+[j] = p,aM,‘f;+[i-11 + ;lAu;+kll + h;+ij-21. (18b)

Again, the interrelationship is more easily seen for the linear system equations, which are

u;+l = fllBMjl(p; - K;,u; - K,u;) + OI,~U; + h;S-’ (19a)

UA n+[jl = PlaMnl(P;+[j-ll _ KAU;+Ij-ll _ KABu;+Ij-ll) +aleu;t[j ~11 +h;+[j--2], (19b)

It can be seen that eq. (19a) can be updated independent of eq. (19b). However, for j > 1, eq. (19b) requires the value of uA ntIi--ll for all nodes that are connected to elements in RAB; let these elements be designated by the subscript 2. If uztl is computed first, then the fractional time step values needed to integrate partition B can be obtained by interpolation. We will here consider linear interpolations on the interface so that

&+lil 252 uI +_!&+1. A m A mA

(20)

An E”’ - E partition with interface interpolation does not insure that all of the difference equations are met at all nodes, so the usual stability criteria no longer apply. Thus, it is necessary to examine the implications of the interface interpolations, which will be done later in this paper.

T. Belytschko et al/Mixed methods for time integration 265

3.4. I - I Partition

For an implicit-implicit partition, the equations of motion are of the form

#A n+l = flOAM;lf;+l + h; (214

UB n+l = po&&lf;+l + h;. (2lb)

In this partition, it is impossible to solve either partition first without any assumptions, for both of the above equations are interrelated. Thus, if the two partitions are to be solved independently, one must be designated as the master and solved first. Let us consider the master domain here to be B; the nodes 2 must be extrapolated. We will here consider extrapolations of the form

%+’ = (1 - cz)$ + czua-‘. (22)

Then eq. (21b) can be solved for ug+l. Once ug+l is obtained, eq. (2 la) can be solved for ui+l, where the latter includes new values at the nodes A.

To illustrate this for a linear system, the equations for partition B are

-1

[p;” - K;B~;+l] ++-MBh; OB

n+1 = u A +-MA +K, -l [&+l

OA - KABu;+l ] + j!-M,h;,

OA

(23a)

(23b)

where the values of ui+l required on the right hand side of eq. (23a) are obtained by eq. (22). Again, since these procedures involve the extraneous element of an extrapolation, its stability properties do not correspond to that of implicit integration of the complete domain.

4. Graphical illustration

The essence of the previous developments can be deduced graphically by considering the flow of information in explicit and implicit time integrations. For this purpose, consider a one dimensional mesh of constant strain elements as shown in fig. 2. Information only travels across one element during one time step in an explicit time integration with a lumped mass, as shown in fig. 2, so the displacement at any node at time step IZ + 1 depends only on the displacements at the two adjacent nodes at time step n. In an implicit integration, the displacements in the entire mesh at the two time steps are coupled.

Thus, as can be seen from fig. 2, E - I partitions require no interpolation or extrapolation, for if the explicit time integration is performed first, the solution at the interface node provides a boundary condition for determining the implicit solution.

However, in E” - E partitions, if the domain with the large time step is integrated tirst, the displacements of the interface nodes must be determined to permit time integration with the

266 T. Belytschko et al./Mixed methods for time integration

t t Explicit - Implicit

t Em-E without lntbrpolation

n+l

n

Fig. 2. Flow of information in mesh partitions.

smaller time step. In fact, two approaches can be taken for determining these intermediate values: the equations at the interface node can be integrated with both the large time step and the small time step, or it can be determined by interpolation. The former scheme has significant disadvan- tages whenever the integer ratio between the time steps m is greater than 2, for the determination of the displacements on the interface in the m - 1 subcycles would then require substep integration in a subdomain of m - 1 elements. This is obviously unwieldy in complex two or three dimensional meshes, where this subdomain would have to be separately identified and treated in the computations.

Fig. 2 also depicts the procedure in an I - I partition. Since information is transmitted across the entire partition instantaneously within a time step, extrapolation is required to establish a boundary condition for whichever partition is integrated first. This extrapolated value is not the final value of the displacement at that point but is modified when the other implicit partition is integrated in time.

4. Stability analysis

For purposes of examining the stability of mesh partitions with interpolations and extrapolations, we consider the model problem shown in fig. 3. Although a model problem provides in- sight into the stability of numerical procedures when applied to practical finite element calculations, only negative findings can strictly be applied to practical calculations, i.e. when a procedure is unconditionally unstable in a model problem, it will clearly not work for practical calculations, but the converse is not true. The model problem consists of a stiff spring, which represents the

T. Belytschko et al./Mixed methods for time integration 261

Fig. 3. Model problem.

the structure, an interface spring, and a third spring which represents the medium. We will restrict our attention to the explicit central difference operator and the implicit trapezoidal operator (the latter is well known in structural dynamics as the Newmark P-method with /3 = l/4).

The homogeneous equations for the model problem are

mii, + c(ti, - zi*) + (k, + k,)u, - k,u, = 0 Wa)

mi.i, + c(U, - til) - k,u, + (k2 + k&u, = 0. Wb)

The central difference and trapezoidal operators, are respectively,

“n 1

U -

(U n+l

At2 - 224” + u”-1)

U “n+l + 2in + tin-1 _ 4 cun+l

At2 - 2u” + u”-1).

(25)

(26)

We first consider the stability of an Em - E partition. In this case we let k, = k, and c = 0 and we restrict our attention to m = 2. The integration of the model problem then consists of integrating node 2 with a time step At, interpolating u2 between time steps n and n + 1 to find 12 + l/2, and then integrating node 2 twice with the time step At/2. The central difference operator with time step At/2 is given by

4 ii” = - (u”+r/2 _ 2u” + Un-1/2) At2

and an interpolation of the form

g+lP = (1 _ Q)un+l + aun

is considered, but for the analysis of the model problem we consider only a= l/2. The equations of the model in the order they are executed are then

(27)

(28)

n+1 U2 - 2~; + u;--’ - 4w,u; + 8w, u; = 0, (294

268 T Belytschko et aLlMixed methods for time integration

kiAt2 \Vi = i= 1 to 2.

4m

We now assume the solution in the form

u’I = A,h2”,

u; = A,h’“:

u”+1/2 = A3X2n+1. 1

Substituting the above into eqs. (29), we obtain

X(w, +w2 - 2) -W,X

x2 + 1 -f W2(h2 + 1)

--4h%J, x4 + h2(8w, - 2) + 1

(30)

(311

If the solution u is stable, the roots of the above determinant must lie within the unit circle. This determinant is given by

X8 + h,X6 + h2X4 + hlh2 + 1 = 0, (32)

where

h, = 2 + h + 2u’; - u2,

II, = 2 + 2b + 4awi - 4~: - a2b,

CI = w1 + IV2 - 2,

h = 8\v, - 2.

Since the determination of the conditions under which the roots X lie within the unit circle is quite complex, we use the transformation

(33)

(34)

T. Belytschko et al./Mixed methods for time integration 269

to map the unit circle in the complex X plane to the left hand part of the complex z plane. Eq. (32) is then transformed to

z4(h2 - 24 + 2) +z2(12 - 2~2,) + 2 + 2h, -t h, = 0. (35)

By the Routh-Hurwitz criterion, a polynomial

P

c cizp-i = 0, i=O

co > 0 (36)

has no roots on the right hand complex plane, if and only if, all of the leading principal minors of the matrix C$ are positive where

c; = c2j_i, (37)

CJ=O if 2j-i<O or 2j--i>p.

For eq. (35), this implies that

4~23 + 12w,w,2 + 8w;w, - 28w2,-4Ow,w,-4w;+48w2-16w,-16GO. (38)

The minimum real root of this equation, which is the maximum time step At which satisfies this equation, is plotted as a function of k,/k, in fig. 4; there is another stable domain between the next two roots, but it is of no practical significance.

For comparison, fig. 4 also shows the stability limit for when both nodes are integrated explicitly with the same timpe step At, and for when node 1 is integrated implicitly, node 2 explicitly (E - I partition). For the former

2 At< -,

0 *ax (39)

where

k, + 3k, + [(k, - k2)2 + 4k;l 1’2 o;,, = __

2m . (40)

The E - I partition was analyzed in a manner identical to the E2 - E partition. The resulting stability condition is simply

(41)

270 T. Belyrschko et al./Mixed methods for time integratiott

1.5G D 0

1.25 - A Straight explicit

0 EL-E Partition

t E 1.00 - E -I Partition

,”

0.75 -

;;

0.50 -

0.25 0

I 3

/ I / , 6 9 12 15

k, +‘kz

Fig. 4. Comparison E2 - E and E - I stability limits for model problem with straight explicit time integration.

which agrees with the stability proof of Belytschko and Mullen [ 31 in that the time step is independent of the stiffness of the implicitly integrated subdomain.

A study of fig. 4 shows that for k,/k, in the range of 2 to 10, the E2 - E partition permits a substantial increase in the time step. However, as the ratio of stiffness increases, the E2 - E partition requires a significantly smaller time step than the E -- I partition; in fact, for these larger stiffness ratios, Em - E partitions match the effectiveness of E - I partitions only if r-n > 2.

We were not able to theoretically deduce the effect of (Y on the stability of E2 - E pa~itions, so we performed numerical experiments with the model problem for various values of CL The model was integrated 1 SO time steps with At = 0.3w2 starting with initial unit velocities; the solution amplitude as a function of (Y is shown in fig. 5. Although this is not a typical strong instability, the growth in the solution after 150 time steps is significantly large and would be quite undesirable. Thus only linear interpolation eliminates any spurious amplitude growth in the model problem.

If we consider the model equations, eqs. (24), with trapezoidal integration of node 1, followed by trapezoidal integration of node 2, then an extrapolation of node z.L+l is needed. Consider the extrapolation

n+l= n U2 U2’ (42)

The difference equations for the model problem in the order they are performed and then

(43aj

-u;+r<w, + d) - 2u;w, - u;-I(w* - d) = 0, (43b)

T. Belytschko et al /Mixed methods for time integration 271

0.6 * 0

/ I I I 0.25 0.5 0.75 1.0 Fig. 5. Solution amplitude growth as a function of interpola-

a tion parameter a! for an E2 - E partition on model problem.

where

2Atc kiAt2 d=_--- and wi =-*

m m

In the z-plane, the characteristic equation becomes

z4(h, - h, -t h, + h, - h,) +z3(4h1 - 2h, - 4fa, + 2h,) +z2(6h, - 2Ja, + 6h,)

+ z(4h, + 2h, -4h,-2h,)+(h,+h,+h,+h,+h,f=o,

where

h, =ab+ad+bd+d2,

h,=Lkzb-- lb-- 16b-3wi-k2ad+2bd-32d-d2-4w,d,

h, = 6ab - 32a - 32b f 256 - 7w2 - 2w,d - d2,

h,=ab-w2-ad-bd+2dw,,

h,=4ab- lti- 16b-2ad-2bd+32d+d2+4w2d-5wi,

(441

(45)

(46)

a=w, +w,+4,

b=w,+w,+4.

272 T. Belytschko et al./Mixed methods for time integration

The Routh-Hurwitz criteria require that

Ci > 0 i = 0 to 4, (47a)

c,c,c, - co(q - c;c4 > 0. (47b 1

We first consider the undamped case. In the absence of damping, d = 0, C, = 0 but

for all At and ratios of stiffness. Therefore, the implicit-implicit partition procedure with extrapolation of eq. (42) is unconditionally unstable. It can be inferred from this conclusion that the procedure is also unsuitable for more general meshes. A similar analysis with the extrapolation

also demonstrates unconditional unstability. When damping is introduced into the system at the interface element, then eq. (47a) is auto-

matically met and stability can be achieved by adding a sufficient amount of damping; the necessary amount is indicated in fig. 6 for a fixed ratio of k,/k, and various ratios of k,/k,. As can be

seen, the damping required to stabilize the model decreases as the ratio of stiffness k, to k, in-

creases and also decreases as k,/k, increases. For a given ratio of stiffness, the required damping

also depends on the time step and achieves a maximum at a critical frequency of the system. Fig. 7 shows these maxima for various ratios of stiffness. In a practical calculation, one would always want the damping to exceed this value. Note that the required damping is not large, for when ex-

pressed in terms of a critical damping of the interface element, as shown in fig. 7, the damping seldom exceeds 2% when k,/k, > 5. Thus, when high frequency response components are of

secondary importance, which is often the case in implicit calculations, stabilization through the

introduction of damping in the interface elements is a realistic alternative.

5. Results

For purposes of ascertaining whether the results of the Em - E analysis of the model problem

are applicable to more general meshes, for values of m > 2 and for nonlinear problems, we considered some one dimensional wave propagation problems. The mesh consisted of 10 elements of length 1 .O, followed by 10 elements of length 0.1, followed by 10 elements of length 1 .O, as shown in fig. 8. Linear displacement, constant strain elements were used. The left hand end was free, the right end fixed. The material properties were chosen so that the elastic wave speed is 1.0, the plastic wave speed is 0.5, and the yield stress is 1 .O. A time step of 0.1 was used at all nodes connected with the small elements, nodes 11 to 2 1, and a time step of 0.8 was used at all other nodes. The interpolation was performed at nodes 10 and 22.

In the first case, a stress of -0.1 was applied as a step function in time at the left hand end. Fig. 8 shows the stress time histories in three elements, which are identified by their midpoint


2.4

0 0.2 3.8 7.4 11.0 14.6 18.2

Time Step Ai = At .& Time Step AT=At .&

6 12 18 24 30

Fig. 6a, b. Interface damping required for ratios of kz/k3 = 4 and kz/ks = 10 as function of the time step and the ratios k3/kl.

0.1 ) IO 20 30 -40

Stiffness Ratio k, /ks Fig. 7. Interface damping in model problem required for unconditional stability for I - I partition.

274 T. Belytschko et aLlMixed methods for time integrution

I .o 2.0 3.0 4.0 5.0 Twne

0 Analytic

/ / 3.0 4.0 50

Time

x = 15.5 V

I I / , / I 0 2.0 3.0 4.0 5.0

Time

Fig. 8. Results for elastic rod problem with E - E partition.

x = 4.5

I I 1 / 0 0.8 1.6 2.4 3.2 4.0

Time

x = IO.45

I / , I 0 0.8 1.6 2.4 3.2 4.0

0 I I / / /

0.8 1.6 2.4 3.2 4.0 Time

Fig. 9. Results for eiastic-plastic rod problem with E - E partition.

coordinates. The results are compared to the corresponding analytic solution. Although the finite element results exhibit the usual dispersive and aliasing phenomena, which are amplified by the use of two element sizes, the c~culations are obviously stable and agree with the analytic solution.

The computations were repeated with other interpolations, including 3 point quadratic interpolations. In all cases, the solutions became unstable shortly after the wavefront reached the interface between the large and small elements.

Fig. 9 shows the results for the same problem when a stress of -2.0 is applied at the left end, and compares them with the analytic solution. In this case, the plastic dissipation eliminates a large part of the spurious oscillations, and the agreement between the analytical and computed solutions is quite good. There is no evidence of any instability due to the use of interface interpolations.

Computations of more complexity have been performed, with both E” - E and I - I partitions, but they will not be reported here. None of these computations exhibited any instabilities or


spurious energy growth. These partition techniques can provide significant savings in structure- media problems, and the results and analysis presented here indicate that the procedures are stable. However, more general proofs of stability would be desirable.

Acknowledgment

We would gratefully like to acknowledge the support of the Electric Power Research Institute.

References

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Mixed methods for time integration