Solution of Burgers' equation for large Reynolds number using finite elements with moving nodes

Solution of Burgers’ equation for large Reynolds number using finite elements with moving nodes J. Caldwell and P. Wanless School of Mathematics i’)tze NE1 8ST, UK

otd Statistics, Newcastle-upon-Tytle

Pol.vtechtk,

Newcostle-npon-

A. E. Cook Deparanent of Mathematics, Technical University of Now Scotia, PO Box 1000, Halifax, Nova Scotia, Carlada 8352X4 (Received February 1985; revised August 1986)

Burgers’ equation often arises in the mathematical modelling used to solve problems in fluid dynamics involving turbulence. Numerical difficulties arise in the solution for the case of large Reynolds number. To obtain high accuracy, finite element methods are important. The aim of this paper is to summarize relevant past work and to use a moving node finite element method to obtain a solution of Burgers’ equation under certain prescribed conditions. The results for high Reynolds number are compared with accurate results obtained by other authors.

Keywords: Reynolds

Burgers’ number

Burgers’ equation is well suited to modelling fluid flows as it incorporates directly the interaction between nonlinear convection processes and diffusive viscous processes. In one dimension, the Cole-Hopf procedure transforms Burgers’ equation into the linear heat conduction equation. Thus, Burgers’ equation has often been used as a model equation for comparing the accuracy of different computational algorithms. This aspect of Burgers’ equation is reviewed by Fletcher’ where earlier references may be found. The one-dimensional Burgers’ equation:

au

au

x+“‘\-=Rg

1 Ai .-

(1)

where II = ~(s,t) in some domain and R is the Reynolds number, often arises in the mathematical modelling

equation,

turbulence,

finite

element

method,

used to solve problems in fluid dynamics involving turbulence. Because of slow convergence, the analytical solution can only be computed for limited values of R. In cases of this type it is necessary to resort to numerical techniques. A number of different numerical approaches to equation (1) have been used by researchers. First of all, finite-difference methods have been shown to be successful and accurate for small Reynolds number R. An interesting approach adopted by Caldwell and Wanless? is to use the method of lines involving finite Fourier series. In this method, a set of ordinary differential equations (ODES) is solved for the amplitudes of the sine terms. This provides a neat alternative to the method used by Sincovec and Madsen3 who solve the ODES directly from the discretized equations using a RungeKutta technique.

Appl. Math. Modelling,

1987, Vol. 11, June

211

Solution

of Burger&equation:

Table 7 Comparison and 16 intervals with

X

0 0.05 0.11 0.16 0.22 0.27 0.33 0.38 0.44 0.50 0.55 0.61 0.66 0.72 0.77 0.83 0.88 0.94 1.00

Table giving

Mitchell Griffiths” accurate solution

et al.

of computed results at r= 1.0 using those of Mitchell and Griffiths” Eight intervals t = 0.005 0 0.0420 0.0840 0.1259 0.1677 0.2096 0.2514 0.2933 0.3351 0.3770 0.4190 0.4609 0.5030 0.5449 0.5863 0.6270 0.6665 0.7044 0

’

0 0.0422 0.0844 0.1266 0.1687 0.2108 0.2527 0.2946 0.3362 0.3778 0.4191 0.4601 0.5009 0.5414 0.5816 0.6213 0.6605 0.6992 0

Moving node results at t = 1.0 using eight position, computed solution and gradient

intervals

Node

Solution at this position

Gradient

1 2 3 4 5 6 7 8 9

0 0.594023 0.859723 0.951990 0.982930 0.993255 0.997219 0.999247 1 .oooooo

0 0.4480 0.6460 0.7095 0.7284 0.7193 0.6867 0.6417 0

0.757 0.756 0.714 0.677 0.355 -4.02 -16.8 -58.2 -2050

Table3 position,

Moving computed

Node

Position of node

Solution at this position

Gradient

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 0.348168 0.562973 0.779499 0.879667 0.952820 0.983549 0.993873 0.997419 0.998696 0.999155 0.999386 0.999550 0.999708 0.999819 0.999906 1.0

0 0.2639 0.4246 0.5828 0.6541 0.7049 0.7260 0.7330 0.7354 0.7360 0.7335 0.7208 0.6850 0.5837 0.4299 0.2461 0

0.759 0.754 0.741 0.719 0.702 0.689 0.683 0.673 0.586 -0.325 -20.6 -114 -367 -1010 -1790 -2410 -2720

Appl.

Table 4 Comparison and 16 intervals with

X

Christie, Griffiths Mitchelllo accurate solution

0.50 0.55 0.61 0.66 0.72 0.77 0.83 0.88 0.94 1 .oo

0.5950 0.6560 0.7150 0.7720 0.8260 0.8760 0.9210 0.9590 0.9871 0.0000

Table giving

5 Moving position,

16 intervals t = 0.001

Position of node

212

eight

and

0.1263 0.1684 0.2103 0.2522 0.2939 0.3355 0.3769 0.4182 0.4592 0.5000 0.5404 0.5806 0.6203 0.6595 0.6983 0

2

J. Caldwell

node results at t= 1.0 using solution and gradient

Math.

Modelling,

1987,

16 intervals

Vol.

giving

11, June

of computed results at t = 0.5 using those of reference 10

eight

and Eight intervals t = 0.005

16 intervals t = 0.001

0.5667 0.6592 0.7201 0.7782 0.8321 0.8805 0.9223 0.9552 0.9783 0

0.5956 0.6567 0.7159 0.7726 0.8263 0.8762 0.9211 0.9593 0.9868 0

node results at t= 0.5 using eight computed solution and gradient

intervals

Node

Position of node

Solution at this position

Gradient

1 2 3 4 5 6 7 8 9

0 0.474133 0.790652 0.924181 0.973475 0.990089 0.996134 0.998848 1.0

0 0.5674 0.8908 0.9696 0.9850 0.9646 0.8995 0.7962 0

1.22 1.13 0.786 0.326 -0.152 -5.23 -25.4 -106 -3300

Table 6 position,

Moving computed

Node

Position of node

Solution at this position

Gradient

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 0.356923 0.539073 0.665096 0.776350 0.843239 0.907400 0.960567 0.986294 0.995015 0.997979 0.999071 0.999454 0.999638 0.999808 0.999905 1 .oooooo

0 0.4315 0.6387 0.7710 0.8749 0.9285 0.9698 0.9921 0.9974 0.9979 0.9976 0.9967 0.9880 0.9441 0.7408 0.4390 0.0000

1.22 1.17 1.09 0.997 0.861 0.734 0.537 0.286 0.109 0.002 -0.363 -3.44 -84.6 -510 -2220 -4000 -4960

node results at t= 0.5 using solution and aradient

16 intervals

giving

The Fourier series method has the advantage that. although the coefficients of the harmonics are small for small values of R (say R = I), they become increasingly dominant its R becomes larger. This indicates the development of ;I wave front which restricts the value of R because the number of necessary Fourier terms becomes large. For R > 10. the method is unsuitable and the solutions indicate that a piecewise polynomial approximation (i.e. finite element) could be attempted where

Solution of Burgers’equation: J. Caldwell et al. the size of the element is not chosen to take into account the nature of the solution. Hence. a finite element method is used to solve Burgers’ equation with the aim of ‘chasing the peak’ by altering the element at each stage using information from the previous step-refer to Caldwell et 01.~The finite element method is extended to the general case of /I elements by Caldwell and Smith” for the case of fixed nodes. The main reasons for considering a finite element approach are: l it is a relatively simple method to formulate l the possibility exists of choosing the size of the elements appropriate to the problem 0 it is able to handle a non-uniform mesh much more readily than finite-difference methods. A number of other authors have used finite elements with moving nodes to solve Burgers’ equation. In particular, Gelinas et al.” have presented a node moving finite element approach which can be applied to large gradients or shocks with high resolution and accuracy. In their system the nodes migrate continuously and systematically to those positions where they are most needed. Also Miller and Miller’ and Miller’ suggest finite element methods which allow many nodes to automatically concentrate in the critical regions and move with them. There are two basic strategies in adaptive finite element constructions. The adaption is made either by introducing additional elements and, therefore, nodes. into the solution domain or by moving the nodes from the initial mesh configuration to regions where the residuals, or some measure of error. is relatively high. These two strategies can be combined. In this paper, the method of nodal movement is developed by using an algorithm which is a generalization of that considered by Caldwell et al.’ Further details of this node-moving approach are given by Cook and Duncan.Y The emphasis in this paper, however, is to produce accurate solutions to Burgers’ equation for large values of the Reynolds number. In order to compare with previous results, the equation is solved in an open rectangle defined by 0 < s < 1, I > 0, under the boundary and initial conditions: u(0.t) = u( 1,r) = 0 I> 0 u(s,O) = sin 7r.v Odsd 1

I(X)

-

with moving nodes

U,(-v)

k

+

ur+I- ur=---I dW,+, k

and hence:

R ch’

U-r+’ dUr+, ct\-

U,,(x)= sin 7rx

0d sd 1

(4) Clearly, equation (3) is nonlinear and any numerical scheme will require iteration or linearization. Previously, Caldwell et ~1.~used a simple linearization technique, namely: u,+,v+, -w*+,u: + UJJ:,,) which proved adequate for values of R up to 100. For larger values of R (e.g. up to 10000) iteration provides a better approximation and this is the technique adopted here. On using the notation:

V solution V, of equation (3) at time t = rk (r = 0, 1, 2, . ..) Vjth estimate of V,, , W (j + 1)th estimate of V,, l equation (3) becomes: W”-RjlW’++

0 < x < 1

(5)

where: W(O)=W(I)=O Equation (5) has the form: o<.u< 1 L(Y) = .f y(O)=y(1)=0 with the operator L and functionfdefined

(6)

by: (7)

R; =

{L(Y) -f}%u

where Y is an approximation to y. Since L is of order two. Y and Y’ must be continuous at the element boundaries. The piecewise cubic interpolant defining the approximation Y over the element meets these conditions and takes the form: Vi) = N, Y; + N,Y; + N,Y,+, + N.,Y,f+, (9) where: N,=l-32’+223

O(k)

Burgers’ equation (1) can be approximated

r> I

(2)

U(s,r,) = U(s,rk) = U,(s) and the approximation : u,,

U,(O)= U,(l)=0

The element residual is defined by:

The results are compared with accurate results obtained by Christie er al.“’ and Mitchell and Griffiths.” Finite element approach Using the notation:

where ’ denotes d/dr. As in previous work, equation (3) will be solved under the boundary and initial conditions:

by:

N2 = C(Z - 22? + Zz) N3 = 32’ - 2Z3 NJ = C( -S + 29) (10) The element length C and local variable Z are given by:

Appl. Math. Modelling,

1987, Vol. 11, June

213

Solution

of Burgers’equation:

C=Xj+l -Xi

J. Caldwell

et al.

z = (A-- x;)/C

(11)

The nodal parameters Yi. Y,!, Y,+ , and Y,!+, are required to minimize Ri for the given choice of nodal position Xi as previously described by Caldwell ef 01.~ On solving for the nodal parameters, the residuals Ri can be computed and used in the node-moving algorithm to update the nodal positions. The movement of the nodes is arranged to satisfy the following four conditions: If residuals Rj and RimI are equal, then node i is not moved. If Rj and Ri_, differ, then node i is moved towards the common element having the larger residual. The order of the nodes must not change. The steady state nodal positions must be asymptotically stable.

Results In the past, Caldwell er al.“ have solved Burgers’ equation under the boundary and initial conditions given in equation (2) using both fixed and moving nodes. The moving node approach was basic and restricted to two elements and although 20 elements were used in the fixed node approach. results were only obtained up to r = 0.25. It is important, therefore, to use the moving node scheme described to obtain accurate results for larger time f, particularly for large values of the Reynolds number. Although most methods fail for very large R, accurate results are available from Mitchell and Griffiths” and Christie et al. I” Computed results have been obtained using the above scheme for R = 10000. Using eight intervals (seven nodes) in the s-direction and taking time interval k = 6r = 0.005, the results obtained at t = 1.0 were found to be close to the accurate solution given by Mitchell and Griffiths” as can be seen from Table I. The results for 16 intervals (15 nodes) with time interval k reduced to 0.001 were even better and again are presented in Table I. Tables 2 and 3 show the position of the nodes and the computed solutions and gradients at these nodes for eight and 16 intervals, respectively. These clearly show how the nodes have moved towards the regions of steepest gradient and gradient change.

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Math. Modelling,

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A further set of results was obtained for t = 0.5 and these are compared in Table 4 with those obtained by Christie et nl.“’ Again, good agreement is found in the case of eight intervals (seven nodes) with improved agreement for the case of 16 intervals (15 nodes). For completeness, Tables 5 and 6 show positions of the nodes and the corresponding solutions and gradients. It is interesting to note how large the values of the gradient W’ are at the boundary x = 1, namely -3300 and -4960 for the eight and 16 interval cases, respectively.

Conclusions Clearly, this simple node-moving algorithms is effective in the solution of Burgers’ equation for large Reynolds number. The same method could be applied to other similar problems and is capable of further extensions.

Acknowledgements The authors would like to thank Dr R. Saunders (Manchester Polytechnic) for both his help and interest in this work.

References I tions

2

Fletcher. C.A.J. Numerical solutions of partial differential equa(ed. J. Noye), North-Holland, Anzsterdatn, 1982. pp. 139-225 Caldwell, J. and Wanless, P. J. P/I~. A: Murh. Gem. 1981. 14,

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P. and Cook, A.E. A linitc clement Appl. Math. Modelling, 1981, S(3).

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Modellitlg. 1982. 6(6). 463468 IO Christie, I. etal. Univ. Dwulee. Mnrhs. Rep. NAI42. IYBO I I Mitchell. A.R. and Griffiiths, D.F. The linite difference method in partial differential equations. Wiley. Chichesrer. IY80