In their method, to solve a one—dimensional moving boundary problem, Crank and Gupta suggest a grid system which moves with the Interface. The method requires some interpolations to be carried out which they perform by using a cubic spline or an ordinary polynomial. In the present paper these interpolations are avoided by employing a Taylor's expansion in space and time dimensions. A practical diffusion problem is solved and the results are compared with those obtained from other methods

Gupta, R

English

Brunel University Research Archive

                        TR/32 OCTOBER 1973 MOVING GRID METHOD WITHOUT INTERPOLATIONS RADHEY S. GUPTA.             A B S T R A C T  In their method, to solve a one—dimensional moving boundary problem, Crank and Gupta suggest a grid system which moves with the Interface. The method requires some interpolations to be carried out which they perform by using a cubic spline or an ordinary polynomial. In the present paper these interpolations are avoided by employing a Taylor's expansion in space and time dimensions. A practical diffusion problem is solved and the results are compared with those obtained from other methods. 1. MOVING GRID METHOD WITHOUT INTERPOLATIONS RADHEY S. GUPTA (Department of Mathematics, Brunei University, Uxbridge, Middlesex, U.K.) 1. Introduction As there is no exact analytical solution available for a general moving boundary problem, various methods have been put forward from time to time. Goodman [1 ] suggests what he calls an 'Integral Method' to get an approximate analytical solution, analagous to the momentum integral used in the field of boundary layer theory in fluid mechanics. Boley [2] gives a 'self-embedding' technique and arrives at two integro- differential equations to be solved usually by numerical methods. landau [3] fixes the moving boundary by making suitable change in the space variable. Chernous'ko [V] introduced an idea of 'isotherms' moving in space with respeot to time while Dix and Cizek [5] have studied this aspect more thoroughly.They transform the basic differential equation so as to make the space variable as a dependent variable and call this method an 'Isotherm Migration Method'. Purely numerical techniques have been given by several authors. Crank [6] suggested the use of Lagrange's formula to deal with unequal intervals near the interface which has been recently employed with some modifications in [7]. Ehrlich [8] uses the implicit scheme by expanding the dependent variable by Taylor's formula in space and time directions near the moving boundary. The details of the various methods to deal with the moving boundary problems in heat flow and diffusion have been given by Muehlbauer and Sunderland [9] and Bankoff [10]. It has to be noted that in all the numerical computations carried out in the foregoing methods the step sizes are kept constant throughout. 2. There are however a few methods which make use of variable grids. Douglas and Gallie [11] divide the whole region into a fixed number of intervals and this size of space mesh is kept fixed for all times but choose each time step such that the boundary moves one space mesh during that time. Murray and Landis [12] use a variable space mesh choosing a fixed time step* They keep the number of space intervals fixed in the region(s) at all times and thus the site of the space mesh either increases or decreases with time. A new technique has been given by Crank and Gupta [13] in which the whole grid is pushed along with the moving boundary, so that the unequal interval is transferred from the neighbourhood of the moving boundary to the fixed surface. It has been shown that this technique improves the degree of smoothness in the motion of the boundary, calculated by the usual method [7] which makes use of a unequal interval near the moving boundary.   In an earlier paper [13] the concept of a Moving Grid System was    introduced and two methods were presented which required interpolations    to be carried at each time step. We now adopt another approach baaed    on the moving grid system but which avoids interpolations. A problem    of biomechanics has been solved and the results have been compared with    those obtained from the method of Murray and Land is [12] and with those    of [7]. 2. Comparative Grid Systems Three different grid systems are shown in figures 1,2 and 3 for a general two-phase problem. The dotted line shows the position of the moving boundary or Interface which divides the two separate regions. In the fixed grid system fig.1 the use of Lagrange formula is made at the mesh point nearest to the moving boundary. We will call this method of computation in future discussions as Fixed Grid Lagrange (FGL) method. As soon as the moving boundary comes too close to the neighbouring mesh 3.  jΔt (j+1)ΔFig.1 Fixed Grid System jΔt  Fig.1 Murray and Landis System of Grid jΔt  (j+1)Δt  x=o x=1 Fig.3 Moving Grid System x=1x=o x=o x=1(j+1)Δt 4. point the process is transferred to the next neighbouring point. At this change over a roughness in the boundary/time graph is observed which disappears in the method under discussion. In the method of Murray and Landis, henceforth called ML method, the space grid size decreases in one region and increases in the other (fig.2). In the system of moving grid the size of the space mesh remains fixed at all times except the intervals nearest to the fixed boundaries, (fig.3). The present method will be called Moving Grid or MG method. It will be seen that for the present problem we need to consider one region only. 3. Statement of the Problem We shall introduce the new method by referring to a practical problem, arising from the diffusion of oxygen in absorbing tissue, which has been described in detail in [7]. Expressed in non- dimensional form we require the solution of the equation,  )1.3(,)(,12txoxutu δ≤≤=∂∂=∂∂  with the boundary conditions ,,, otoxotu ≥==∂∂ (3.2) ,,)(, ottxoxuu ≥==∂∂= δ (3.3) and the initial condition u = ½ (1-x)2 , o ≤ x ≤ 1 , t = o , (3.4.) 5. where S(t) denotes the position, of the moving boundary at time t. 4. Position of the Moving Boundary By differentiating u with respect to t and using basic equation (3.1) it is easy to show that, (4.1) Similarly it can also be shown that at the moving boundary x = δ  (4.2) where  and  denotes the first and second derivatives of 8 with respect δ& δ&&to t. Further if h is the distance of a point where u = ur from the moving boundary, in its neighbourhood, where we haveu 0xuu =∂∂=  it is easy to show by using (4.1) and (4.2) in the Taylor's series for ur that if the boundary is not moving too quickly then, .)ur2(~h 21−(4.3) 5. Description of Method          We subdivide the whole region o ≤ x ≤ 1 into n intervals each of width Ax such that xi = iΔx ; i = , 1 ,..., n (nΔx = 1) at t =  o.  As the boundary moves a distance at the next time step At the whole grid system is moved a distance ∈  towards the fixed surface x = o. The size of the first interval will then reduce to Δx - ∈ = ξ1 (say) and the new mesh points at t = Δt will be x1 - ∈, x2 - ∈, ... etc. In general let us suppose that the position of the ith mesh point at ).(,122txxu δ==∂∂  .,,3...352.44.33etcxuxuxu δδδδ −−=∂∂=∂∂−=∂∂  6. t = jΔt is denoted by xji then following relations hold, xji = ξJ + (i-1) Δx , i = 1, 2, .. , (5.1) xj+1 - Xji - ∈j+1 , (5.2) ξj+1 = ξj - ∈j+1 (5.3) where ∈j+1 is the distance traversed by the moving point from time jΔt to (j+1) Δt, If hj+1 denotes the value of h at the (j+l)th time step then, ∈j+1 = ΔX- hj+1. (5.4) Let us now write the Taylor's series for u in two variables (space and time) as follows.  ⎭⎬⎫⎩⎨⎧∂∂∈−∂∂Δ+=Δ+∈−tututtxuttxu ),(),(   ......2212222222 +⎭⎬⎫⎩⎨⎧∂∂∈+∂∂∂Δ∈−∂∂Δ+xutxuttut  where ∈ and At are as defined before. from (3.1) the above formula may be written as given below, +∂∂Δ+∂∂Δ∈−∂∂∈+⎟⎟⎠⎞⎜⎜⎝⎛ −∂∂Δ+∂∂∈−=Δ+∈− 2223322222221),(),(tutxutxuxutxutxuttxu               (5.5) puttingtxuandtu∂∂∂∂∂ 2  7.  truncated form, (5.6) Formula (5.6) may give the value of u at the next time step of Δt at a grid point which has moved a distance ∈ during that time. By comparison the governing differential equation in the ML method is written as follows, (5.7) where (5.8) Equation (5.7) may be rewritten using (5.8) and the basic equation (3.1) to give, (5.9) where δ (t) is the position of the moving boundary at time t. In order to compare results obtained from the MG method and with those from the ML method we ignore the last term in (5.6) and write Again neglecting 2233tuandxu∂∂∂∂ we werite (5.5) in the following .21),(),( 22222xuxutxutxuttxu ∂∂∈+⎟⎟⎠⎞⎜⎜⎝⎛ −∂∂Δ+∂∂∈−=Δ+∈−  tudtdxxudtdu∂∂+∂∂=  ..δδxdtdx =   ,1, 22.⎟⎟⎠⎞⎜⎜⎝⎛ −∂∂+∂∂=xuxxudtdu δδ  8. it in the following finite-difference form,   (5.10)         ui, j+1 = ui,j - ∈j+1 ui,jΔt (ui,j-1)  where dashes show the order of differentiation with respect to x. It should be remembered that ui,j is different in the present paper than used in the ordinary sense as xi is not fixed and varies with time, The values of ui and ui, at anr time, may be written using the usual finite difference formulae, xuuiu ii Δ−= −+211'(5.11) .2211'xuuuiu iii Δ+−= −+(5.12) i = 2, 3, ... At the grid point x = x1 Lagrange type formulae have to be used allowing for the unequal interval ξ nearest to the surface x = o. The respective equations would be, and)x(xux)x()x(xuu o21 ξΔ+ξΔ−Δξξ−Δ+Δ+ξΔξ= (5.13) .)x(uxu)x(xu2u o12'1 ⎭⎬⎫⎩⎨⎧Δ+ξξ+Δξ−Δ+ξΔ= (5.14) 9. Let us assume that the function values ui,j , i=o, 1, ... r, (r+1) are known at time jΔt when the distance of the moving boundary from the surface x=o is ξJ + rΔx. The value of ur,j+1 is found using an explicit finite difference formula from the basic equation (3.1). Once ur is known the value of h is known from (4.3) and hence ∈j+1 is known from (5.4). The values of ui,j+1 , 1 = 1 , 2 ,  ... r are calculated from (5.10) using (5.11) through (5.14). The relations similar to (5.10), (5.11) and (5.12) maybe written for the ML method and the values of u may be computed in exactly the same manner as described above. The essential difference between the ML and the MG method is that the movement of each grid point in the present method is the same as that of the moving boundary while in the ML method it is proportional to its distance from the fixed surface x=o. The width of the space meshes except the one nearest to the surface remains constant in the MG method whereas it goes on decreasing in the ML method. 6. Results and Discussion We notice that there is a discontinuity in the surface-gradient at t=o. Because of this the numerical methods based on finite differences are liable to give inaccurate solutions in the neighbourhood of the surface for short times. In an earlier paper [7], however, an analytical solution satisfactory for small times was obtained which is given by        )1.6(t2xxerfot2xexpxt2)x1(21)t,x(u22 ⎟⎟⎠⎞⎜⎜⎝⎛+⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛−−=      o ≤ x ≤ 1 and t small. 10. We start the present solutions from the values taken from (6.1) at t=0. 025 when the boundary δ  = 1 , has not moved to an accuracy of six significant figures based on the FGL method. The positions of the moving boundary and the surface values of u are computed in Tables I and II respectively for the following methods: (i) Forward Difference Lagrange (FGL) method, (ii) Moving Grid (MG) method and (iii) Murray and Landis (ML) method. The agreement between various results seem to be very good. The results obtained in Table I from the MG method are very much nearer to the results obtained from the FGL method than those obtained from the ML method. In Table II all the results are almost identical. Table III gives the positions of the moving boundary at and around the times when the process for calculating u, in the neighbourhood of the moving point, in the FGL method, is transferred one space interval towards the surface x=o. The corresponding figures are given for the MG method. The irregularities produced in the former method are clearly visible, whereas their counterparts show a smooth behaviour throughout. Table IV gives the surface values of u computed from the present method at and around such times when the first space interval ξ is increased to ξ+Δx for succeeding computations. It is seen that the differences in the values of u show no sign of irregularities. The comparative figures for the FGL method are also given. I am extremely grateful to Professor J. Crank, Head of the School of Mathematical Studies, Brunei University, Uxbridge, U.K., for many helpful discussions. 11. TABLE I Comparison of 104  at different times. The numerical δsolutions start from the analytical solution at t = 0.025.              Time  Method          0.040   0.060   0.100 0.120  0.140   0.160 0.180 0.185FGL Ax=0.10  9988  9905  9312 8747 7912  6756 4849 4014MG  Δx=0.10  9988  9903  9301 8719 7885  6682 4766 4048ML  9988  9904  9309 8740 7930  6776 4974 4308NOTE: The ML method has been started from Δx = 0,10. TABLE II Comparison of 104u at the surface at different times. All solutions start from analytical solution at t = 0.025        TIME  Method    0.040   0.060   0.100   0.120   0.140   0.160   0.180  0.185 FGL Ax=0.10 2745 2238 1434 1094 781 490 220 156MG  Δx=0. 10 2745 2238 1434 1093 780 490 219 155ML  2745 2238  1434   1093 780 489 218 154NOTE: The ML method has been started from Δx = 0.10. 12. TABLE III Table showing the irregularities in the position of the moving boundary, calculated by the PGL method. Comparatively smooth figures are shown for the MG method (Δx = 0.10) FGL Method  MG Method      Time 104δ - ∆ - ∆2 104 δ - ∆ -∆z  0.110 9099 9070 9040 9010 8984 29 30 30 26  1  0 -4 9094 9066 9037 9008 8978 28 29 29 30 1 0 2 0,137 8141 8089 8034 7994 7954 52 55 40 40   3 -15   0 8122 8076 8030 7982 7934 46 46 48 48 0 2 0 0.154 7277 7204 7124 7037 6985 73 80 87 52   7   7 -35 7220 7157 7094 7028 6962 63 63 66 66 0 3 0 0.167 6396 6306 6203 6045 5979 90 103 158 66  13 55 -92 6294 6211 6126 6038 5949 83 85 88 89 2 3 1 0.176 5499 5393 5268 5020 4937 106 125 248  83   19  123 -165 5461 5355 5245 5132 5014 106 110 113 118 4 3   5 0.184 4652 4538 4406 4014 3912 114 132 392 102   18  260 -290 4498 4355 4205 4048 3883 143 150 157 165 7 7 8 NOTE:  The data are tabulated at an interval of time Δt = 0.001, The underlined values correspond to the times, shown in column 1, when the interpolation process near the moving boundary in FGL method is transferred one step towards the fixed surface x = 0. 13. TABLE IV Table showing the smoothness of the function values at the fixed surfaoe calculated by the MG method. Comparative figures are shown for the FGL method (Δx = 0.10)  Time FGL Method MG Method   104u - A 104u - A .093 1598 1580 1561 1543 1524 18 19 18 19 1598 1580 1561 1543 1525  18 19 18 18 .127 1013 997 981 965 950 16 16 16 15 1012 996 981 965 949  16 15 16 16 .148 691 677 662 647 633 14 15 15 14 691 676 661 647 632  15 15 14 15 .163 476 462 448 435 421 14 14 13 14 476 462 448 434 420  14 14 14 14 .174 326 312 299 286 272 14 13 13 14 325 312 298 285 272  13 14 13 13 182 220 207 194. 181 168 13 14 13 13 219 206 194 181 168  13 12 13 13 .188 143 130 117   105  92     13          13           12         12      142 130 117 105 92   12 13 12 13NOTE: The data are tabulated at an interval of time Δt = 0,001. The underlined values correspond to the times, shown in column 1 . when the first space interval, in MG method, is increased by Δx. 14. REFERENCES 1. Goodman, T.R. , AS ME Transactions, 80 (1958) pp.335-342. 2. Boley, B.S. , J. Maths. and Phys., 40 (1961) pp.300-313. 3. Landau, H.G., Quart. Appld. Maths., 8 (1950) pp. 81-94. 4. Chernous'ko, F.L. , Int. Chem. Engg. , 10 (1970) pp. 42-48. 5. Dix, R.C. and Cizek, J. , Proc. Int. Heat Transfer Conference,    Paris, 1971 , Vol.1. 6. Crank, J., Quart, J. Mech. Appld. Maths., 10 (1957) pp.220-231. 7. Crank, J. and Gupta, R.S. , J. Inst. Maths. Applies., 10 (1972) pp.19-33. 8. Ehrlich, U., J. ACM., 5 (1958) pp.161-176. 9. Muehlbauer, J.C. and Sunderland, J.E. , Appl. Mech. Rev. 8 (1965) PP. 951-959.  10. Bankoff, S.G. , Advances in Chemical Engineering, Vol.5, edited by Drew, Hoopes and Vermeulen, Academic Press, 1964, pp.75-150. 11. Douglas, J. and Gallie, T.M., Duke Math. J. 22 (1955) PP.557-570. 12. Murray, W.D. and Landis, P., J. Heat Transfer, 81 (1959) pp. 106-112. 13. Crank, J. and Gupta, R.S., J. Inst. Maths. Applies., 10 (1972) pp.296-304. 

Advances in Chemical Engineering, Vol.5, edited by Drew, Hoopes and Vermeulen,

Moving grid method without interpolations

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Moving grid method without interpolations

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