A simple, modified finite-difference method is described for solving Laplace's equation with boundary singularities of the infinite derivative type. Modified approximations for the derivatives of the Laplacian equation are employed near the singularity. These are developed from a truncated series form of the local analytical solution. The method is applied to the problem of Motz. The numerical results compare favourably with those obtained by other techniques

Crank, J

Furzeland, RM

Romanian

Brunel University Research Archive

                  TR/60     March  1976  MODIFIED FINITE-DIFFERENCE APPROXIMATIONS NEAR THE SINGULARITY IN THE PROBLEM OF MOTZ  by J.  Crank and R.M. Furzeland                   ABSTRACT  A  s i m p l e ,  m o d i f i e d  f i n i t e - d i f f e r e n c e  m e t h o d  i s  d e s c r i b e d  f o r  s o l v i n g  L a p l a c e ' s  e q u a t i o n  w i t h  b o u n d a r y  s i n g u l a r i t i e s  o f  t h e  i n f i n i t e  d e r i v a t i v e  t y p e .    M o d i f i e d  a p p r o x i m a t i o n s  f o r  t h e  d e r i v a t i v e s  o f  t h e  L a p l a c i a n  e q u a t i o n  a r e  e m p l o y e d  n e a r  t h e  s i n g u l a r i t y .  T h e s e  a r e  d e v e l o p e d  f r o m  a  t r u n c a t e d  s e r i e s  f o r m  o f  t h e  l o c a l  a n a l y t i c a l  s o l u t i o n .  T h e  m e t h o d  i s  a p p l i e d  t o  t h e  p r o b l e m  o f  M o t z .  T h e  n u m e r i c a l  r e s u l t s  c o m p a r e  f a v o u r a b l y  w i t h  t h o s e  o b t a i n e d  b y  o t h e r  t e c h n i q u e s .   – 1 –  1 .     I n t r o d u c t i o n   T h e  p r o b l e m  o f  M o t z  [ 5 ] ,  w i t h  i t s  s i n g u l a r i t y  d u e  t o  a  r e - e n t r a n t  c o r n e r  o f  i n t e r n a l  a n g l e  2 π  ( F i g u r e  1 ) ,  h a s  b e e n  t r e a t e d  b y  m a n y  a u t h o r s  t o  d e m o n s t r a t e  t h e  e f f e c t i v e n e s s  o f  t h e i r  s i n g u l a r i t y  t r e a t m e n t s .  A n  a l t e r n a t i v e  f o r m u l a t i o n  o f  t h e  p r o b l e m ,  b a s e d  o n  i t s  a n t i s y m m e t r i c  p r o p e r t i e s ,  w a s  g i v e n  b y  W o o d s  [ 1 5 ]  ( F i g u r e  2 ) ,  a n d  i t  i s  i n  t h i s  f o r m  t h a t  t h e  p r o b l e m  i s  t r e a t e d  i n  t h e  l i t e r a t u r e  [ 4 ,  6 ,  7 ,  9  –  1 4 ] .   I n  t h i s  r e p o r t  t h e  n a t u r e  o f  t h e  s i n g u l a r i t y  i s  d i s c u s s e d  a n d  a  n e w  m e t h o d  p r o p o s e d  f o r  o b t a i n i n g  n u m e r i c a l  s o l u t i o n s  n e a r  t h e  s i n g u l a r i t y .  T h e  p r o p o s e d  m e t h o d  i s  b a s e d  o n  t h a t  u s e d  b y  M o t z  i n  t h a t  i t  u s e s  m o d i f i e d  f i n i t e - d i f f e r e n c e  a p p r o x i m a t i o n s  n e a r  t h e  s i n g u l a r i t y  w h i c h  a r e  d e v e l o p e d  f r o m  a  l o c a l  t r u n c a t e d  s e r i e s  f o r m  o f  t h e  s o l u t i o n .  T h e  m e t h o d  d i f f e r s  i n  t h a t  t h e  m o d i f i e d  a p p r o x i m a t i o n s  a r e  d e v e l o p e d  f r o m  t h e  d e r i v a t i v e s  i n  t h e  g o v e r n i n g  L a p l a c i a n  e q u a t i o n  r a t h e r  t h a n ,  a s  d o n e  b y  M o t z ,  f r o m  a l g e b r a i c  e q u a t i o n s  f o r  t h e  s o l u t i o n  v a l u e s .  T h e  m e t h o d  a l s o  d i f f e r s  i n  t h e  s e t  o f  n e i g h b o u r i n g  p o i n t s  u s e d  t o  a p p r o x i m a t e  t h e  u n k n o w n  c o e f f i c i e n t s  i n  t h e  t r u n c a t e d  s e r i e s  a p p r o x i m a t i o n s  f o r  t h e  d e r i v a t i v e s .  T h e  m e t h o d  i s  a n  e x t e n s i o n  o f  t h e  m e t h o d  o f  B e l l  a n d  C r a n k  [ 3 ] .   –  2  –  T h e  p r o b l e m  o f  M o t z  i s  t o  s o l v e  02y22x2=∂φ∂+∂φ∂       ( 2 . 1 )  i n  t h e  s q u a r e  c o n t a i n i n g  a  s l i t  ( F i g u r e  1 ) .    T h e  s l i t  r e p r e s e n t s  a  r e - e n t r a n t  c o r n e r  o f  i n t e r n a l  a n g l e  2π  w i t h  N e u m a n n  b o u n d a r y  c o n d i t i o n s  o n  t h e  a r m s  OB  a n d   C o n v e r t i n g  t o  l o c a l  p o l a r  .OHc o - o r d i n a t e s    ρ=x c o s θ ,  y  =  ρ s i n θ  ,  c e n t r e d  o n  0 ,    02θφ22ρ1ρφρ12ρφ2 =∂∂+∂∂+∂∂  ,     ( 2 . 2 )  W i t h  0=θ∂φ∂    o n    θ    =    0    a n d    θ    =    .2π   ( 2 . 3 )   M o t z  s o u g h t  t h e  l o c a l ,  v a r i a b l e - s e p a r a b l e  s o l u t i o n  o f  ( 2 . 2 ) ,    φ ( ρ , θ )    =    R ( ρ )ψ ( θ ) ,      ( 2 . 4 )   W h i c h  g a v e     n , A , B , C  a r b i t r a r y  c o n s t a n t s ( 2 . 5 )   ⎪⎭⎪⎬⎫=+=ψnρCRθnsinBθncosA F i t t i n g  t h e  b o u n d a r y  c o n d i t i o n  ( 2 . 3 )  g a v e   ...,2,1,0k,2kn;0B ===  ( 2 . 6 )   T h u s  t h e  l o c a l  s e r i e s  f o r m  o f  t h e  s o l u t i o n  n e a r  t h e  s i n g u l a r i t y  a t  0  i s   ....2θ3cosρ3cθcosρ2c2θcosρ1c0cφ2321++++=   ( 2 . 7 )   w h e r e  t h e   a r e  u n k n o w n  c o n s t a n t s  ( t o  b e  f o u n d ) .ic– 3 –  3 .    A n  a n t i s y m m e t r i c  F o r m u l a t i o n ,  W o o d s   [ 1 5 ]   W o o d s  u s e d  t h e  f a c t  t h a t  φ -  5 0 0  i s  a n t i s y m m e t r i c  a b o u t  t h e  l i n e  B E  c o n t a i n i n g  t h e  s l i t  a n d ,  b y  i m p o s i n g  t h e  b o u n d a r y  c o n d i t i o n  φ  =  5 0 0  o n  E 0 ,  o n l y  n e e d e d  t o  c o n s i d e r  t h e  t o p  h a l f  o f  t h e  s q u a r e  ( F i g u r e  2 ) .  H o w e v e r ,  t h i s  c a n  l e a d  t o  s o m e  c o n f u s i o n  i n  t h a t  t h e  l o c a l  s e r i e s  f o r m  o f  t h e  s o l u t i o n  i s  a l t e r e d  s i n c e  t h e  r e - e n t r a n t  c o r n e r  n o w  h a s  t o  b e  r e g a r d e d  a s  h a v i n g  a n  i n t e r n a l  a n g l e  o f  π  w i t h  b o u n d a r y  c o n d i t i o n s   0θφ =∂∂    o n    θ   =   0   ;   φ   =   5 0 0    o n    θ   =   π  .   ( 3 . 1 )   A p p l i c a t i o n  o f  ( 3 . 1 )  t o  ( 2 . 5 )  g i v e s    B   =   0    ;    n   =   k  +  21  ,     k =  0 , 1 , 2 , . . .   ( 3 . 2 )   T h i s  g i v e s  t h e  n e w  l o c a l  s e r i e s  f o r m          ...25cos3c23cos2c2cos1c0c252321 +θρ+θρ+θρ+=φ    ( 3 . 3 )   C o m p a r i n g  t h i s  w i t h  ( 2 . 7 )  w e  n o t e  t h a t  t h e r e  a r e  n o  l o n g e r  a n y  t e r m s  i n  i n t e g e r  p o w e r s  o f  ρ  a n d  m u l t i p l e s  o f  θ ,  b u t  t h a t  t h e  f i r s t  t w o  t e r m s  a r e  t h e  s a m e .  T h e  d i f f e r e n c e  i n  f o r m s  d i d  n o t  a f f e c t  t h e  w o r k  o f  W o o d s  s i n c e  h e  u s e d  o n l y  t h e  f i r s t  s i n g u l a r  t e r m ,  .   W a i t  a n d  1c– 4 – M i t c h e l l  [ 1 0 ] ,  h o w e v e r ,  u s e d  a n  a s y mp t o t i c  e x p a n s i o n  o f  t h e  f o r m (2 . 7 )  up  t o  power s  o f  23ρ  f o r  t h e  a n t i s y mme t r i c  fo r m o f  t h e  p r o b l e m,  -  s t r i c t l y  s p e a k i n g  t h e y  s h o u l d  h a v e  u s e d  a n  e x p a n s i o n  o f  t h e  f o r m ( 3 . 3 ) .     4 .    S t anda rd  F in i t e -d i f f e r ence  So lu t i on  fo r  Woods '   Fo rm  T h e  p r o b l e m o f  F i g u r e  2  i s  s c a l e d  b y  s e t t i n g   ( i )    u   =   φ   -   5 0 0         ( 4 . 1 )   ( i i )    B   =    ( 1 , 0 )    ;    c  =  ( 1 , 1 )   ;    D  =  ( - 1 , 1 )   ;     E  =  ( - 1 , 0 )  .  ( 4 . 2 )   T o  e n a b l e  c o mpa r i s o n s  w i t h  t h e  w o r k  o f  M o t z  a n d  Wo o d s ,  t h e  r e g i o n  i s  d i s c r e t i s e d  a s  s h o w n  i n  F i g u r e  3  w i t h  δx  =  δy  ≡  h  =  2 / 7 .   T h i s  d i s c r e t i s a t i o n  r e s u l t s  i n  a  me s h  l e n g t h  o f  h / 2  a t  t h e  e d g e s  ED,  DC and  CB and ,  f o l l owing  Motz  and  Woods ,  va lue s  on  ED and  DC a r e  n o t  c o mp u t e d .  I n s p e c t i o n  o f  t he  T a y l o r  s e r i e s  a p p r o x i ma t i o n s  fo r  po in t s  on  t he  me sh  l i ne s  h /2  away  f rom the  edges  ED,  DC and  CB shows t h a t  t h e  d i s c r e t i s a t i o n e r r o r  i s  o n l y  0 ( h )  f o r  t h e s e  p o i n t s .  – 5 –  T h e  g o v e r n i n g  e q u a t i o n  i s   02yu22xu2 =∂∂+∂∂  .        ( 4 . 3 )   A p p r o x i m a t i o n  o f  ( 4 . 3 ) ,  u s i n g  t h e  s t a n d a r d ,  c e n t r a l - d i f f e r e n c e  f o r m u l a e ,  l e a d s  t o  t h e  u s u a l  f i v e - p o i n t  a p p r o x i m a t i o n s  o f  )2h(0   0j,i4u1j,iu1j,iuj ,1iuj,1iu =−++−+++−    ( 4 . 4 )   f o r  p o i n t s  ( i , j )  n o t  i n v o l v i n g  t h e  b o u n d a r y ,  w h e r e    x  =  i h   ,    i  =  0 , 1 , 2 , . . .   ;   y  =  j h   ,   j  =  0 , 1 , 2 ,  . . .  ( 4 . 5 )   F o r  p o i n t s  i n v o l v i n g  t h e  N e u m a n n  b o u n d a r y  c o n d i t i o n  t h e  u s u a l  c e n t r a l  d i f f e r e n c e  a p p r o x i m a t i o n s  f o r  t h e  d e r i v a t i v e s  a r e  u s e d  a n d  t h e  f i c t i t i o u s  p o i n t  a r g u m e n t  a p p l i e d .  F i g u r e  8  c o m p a r e s  t h e  n u m e r i c a l  s o l u t i o n  o b t a i n e d  u s i n g  t h e s e  s t a n d a r d  f i n i t e -  d i f f e r e n c e  a p p r o x i m a t i o n s  w i t h  t h e  h i g h l y  a c c u r a t e  r e s u l t s  p r o d u c e d  b y  t h e  c o n f o r m a l  t r a n s f o r m a t i o n  m e t h o d  o f  P a p a m i c h a e l  a n d  W h i t e m a n  [ 7 ] .   T h e  r e s u l t s  s h o w  t h a t  a  h i g h  d e g r e e  o f  i n a c c u r a c y  o c c u r s  n e a r  t h e  s i n g u l a r i t y  a n d  i l l u s t r a t e  t h e  f a c t  t h a t  i n a c c u r a c i e s  s p r e a d  t h r o u g h o u t  t h e  e n t i r e  r e g i o n .  T h i s  i s  c a l l e d  t h e  p o l l u t i o n  e f f e c t  b y  B a b u s k a   a n d  A z i z  [ 1 ] .    5 .     M o d i f i e d  F i v e - p o i n t  A p p r o x i m a t i o n s  n e a r  t h e  S i n g u l a r i t y   I n s t e a d  o f  a p p l y i n g  t h e  s t a n d a r d  f i n i t e - d i f f e r e n c e  a p p r o x i m a t i o n s  t h r o u g h o u t  t h e  e n t i r e  r e g i o n ,  a  n e i g h b o u r h o o d  N ( 0 )  n e a r  t h e  s i n g u l a r i t y  i s  c h o s e n  a n d ,  f o r  p o i n t s  i n  N ( 0 ) ,  m o d i f i e d  a p p r o x i m a t i o n s  a r e  d e v e l o p e d  w h i c h  t a k e  i n t o  a c c o u n t  t h e  n a t u r e  o f  t h e  s i n g u l a r i t y .  T h e s e  m o d i f i e d  a p p r o x i m a t i o n s  a r e  f o r m e d  b y  a p p r o x i m a t i n g  e a c h  o f  t h e  d e r i v a t i v e s  i n  ( 4 . 3 )  u s i n g  t h e  l o c a l  p o l a r  c o - o r d i n a t e  s e r i e s  –  6  – form  ...250cos3c23cos2c2cos1c),(u252321 +ρ+θρ+θρ=θρ    (5.1)  which is obtained by comparison with (3.3).   A five-point approximation is  formed by taking the following three-term truncated form of (5.1) :   .θ21)(2icos212iρθ)(ρ31iifwhereθ)(ρificθ)(ρu−−=∑==∗ ,,,    (5.2)  Using the standard differential relations  ,u22sinu2sin2u222sinu22sin2u22cos2xu2θ∂∂ρθ+ρ∂∂ρθ+θ∂∂ρθ+θ∂ρ∂∂ρθ−ρ∂∂θ=∂∂   (5.3)  ,u22sinu2cos2u222cosu22sin2u22sin2yu2θ∂∂ρθ+ρ∂∂ρθ+θ∂∂ρθ+θ∂ρ∂∂ρθ−ρ∂∂θ=∂∂   (5.4)  where approximations for the ρ  and θ  derivatives are obtained by differentiating (5.2),  the following three-term series approximations  for 2222yu,xu∂∂∂∂ are found:  ,)(0errortruncationwith)6.5(),(31i'iwic2yu2)5.5(),(31iiwic2xu223ρ⎪⎪⎭⎪⎪⎬⎫θρ∑==∂∗∂θρ∑==∂∗∂     ⎪⎪⎭⎪⎪⎬⎫=−=θρ=ρθ=ρθ−=.3,2,1i,iw'iw42/cos15w;42/cos3w;42/3coswwhere212123 321   (5.7)     The symmetry of the  arises from the symmetry of (5.3) and iw(5.4). - 7 -  Approximations for the constants , i = 1,2,3 are found in terms icof neighbouring u values by using the three-term approximation (5.2). Referring to Figure 4, and denoting j,j,ju θρ∗  to be the corresponding u*, ρ,θ, values at the points j = 1,2,3,4,5, then u* values at three neighbouring points in the horizontal direction are used to find the 1intpo2x*u2for)5.5(inic∂∂   The most obvious choice, and that which leads to a five-point formula, is to use points j = 1,2 and 3 to give the following three equations for the , ic∑= =θρ=31i.3,2,1j,)j,j(ificj*u     (5.8)  The solution of (5.8) is denoted by .3,2,1i,*3uiD*1uiC*2uiBic =++=     (5.9) Similarly, to find the  in (5.6) for ic 1intpo2y*u2∂∂, points 4,1 and 5 are used in the vertical direction.  The solution is denoted by .3,2,1i,*5u'iD*1u'iC*4u'iBic =++=     (5.10) - 8 -  Substitution of (5.9) and (5.10) in (5.5) and (5.6) respectively gives the five-point approximation   ,0*1u1e*5u5e*4u4e*3u3e*2u2e2y*u22x*u2*u2 =++++=∂∂∂∂=∇   (5.11) ⎪⎪⎭⎪⎪⎬⎫∑= ∑= +==∑= ∑= ∑====31i31i,'iC'iw'iC'iw1e,'iD'iw5e31i31i31i,'iB'iw4e,iDiw3e,iBiw2ewhete    (5.12) for a typical point 1 in N(0) , the  and  being evaluated iw 'iwat the point 1.  The above technique of horizontal and vertical derivative replacement is based on that used by Bell and Crank [3] who treated 2yu2∂∂  as above but used the standard central difference formula for 2xu2∂∂  .  The method used above is a generalisation of the ideas of Bell and Crank in that    (i)   both derivatives are treated   (ii)   the approximations (5.5) and (5.6) are written in a general way so that the neighbouring points chosen need not lie on the same horizontal or vertical line.  The generalisation (ii)  is useful in developing higher-order,  multi-point modified approximations by varying the number of terms included in the truncated series expression and the set of neighbouring points used for any one point in N(0).  The method can be extended to the time-dependent case by following Bell and Crank .  The neighbourhood N(0) can include points away from the singularity at 0, as long as the three-term approximation (5.2) remains valid. - 9 - This may be checked as described in Motz [5].   The approximate s of N(0) can be determined by noting that the discretisation error in the standard finite-difference approximations (4.4) is 0(h2) whereas the modified approximations (5.5) and (5.6) contain a truncation error of )(0 23ρ .  Thus application of the modified approximations is advantageous as long as the truncation error does not exceed the discretisation error.  An approximate rule is then to choose N(0) such that the maximum ρ value in N(0),  .maxρsay, is such that 23.maxρ  is of the same order of magnitude as h2, - practical experience suggests that 23.maxρ  < 5h2 is a useful guide.  In practice only a few points in N(0) are needed.  Five-point 'molecules' differing from that given in Figure 4 are needed for points in N(0) which involve the boundary.  Points to the right of 0 on j  = 0 do not have a point at the j-1 level,  necessitating a different molecule,  e.g.  Figure 5,   Points to the left  of 0 on j  = 1 involve points on j  = 0 for which θ  = π .   The fact that θ  = π  means that each of the ),(if θρ  in (5.2) are zero, and thus solutions to (5.8) cannot be found.  A suggested alternative is  given in Figure 6.   The first  point on the right of 0 on j  = 0 involves both the above problems and a suggested molecule is given in Figure 7.  The general form of the modified approximations allows for any combination of five neighbouring points provided θ ≠ π  .  - 10 –  6.   Numerical Results  Referring to Figure 9, the four immediately neighbouring points   around 0 are chosen for N(0)  since here 4321 PandP,P,P23.maxρ  ≈  0.26 and h2 ≈  0.08.   This choice of points is  similar to that used by Motz  and Woods and enables comparisons to be made with their results.   Modified approximations of the form (5.11) are applied at points inside N(0) using the suggested molecules of Figures 5 - 7 where appropriate.  Standard finite-difference approximations (4.4) are used for points outside N(0).  The highly accurate values of Papamichael and Whiteman [7] based on a conformal transformation are used to represent exact values. The results obtained are comparable with those of Motz and Woods and give good agreement with the conformal transformation values.  The values of the  in (5.11) are given in Figures 10-13 for iethe points  with  h = 2/7.  The values given have been scaled 41 PP −so that = -4.0000. ie               - 14 - References 1.    Babuska, I. and Aziz, A.K., The finite element method for non-smooth domains and coefficients, pp. 265-283 of A.K. Aziz (ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial  Differential  Equations. Academic Press, New York, 1972.  2.    Bell, G.E. and Crank, J., A method of treating boundary singularities in time-dependent problems.  J. Inst. Maths. Applies. 12, 37-48, 1973.  3.    Bell, G.E. and Crank, J., A simple finite-difference modification for improving accuracy near a corner in heat flow problems.  Technical Report TR/51, Department of Mathematics, Brunel University, 1975.  To appear in Int. J. Num. Meth. Engng.  4.    Gregory, J.A. and Whiteman, J.R., Local mesh refinement with finite elements for elliptic problems.  Technical Report TR/24, Department of Mathematics, Brunel University, 1974.  5.    Motz, H., The treatment of singularities of partial differential equations by relaxation methods. Quart. Appl. Math. 4, 371-377, 1946.  6.    Papamichael, N. and Rosser, J.B., A power series solution for a harmonic mixed boundary value problem. Technical Report TR/35, Department of Mathematics, Brunel University, 1973.  7.    Papamichael, N. and Whiteman, J.R., Treatment of harmonic mixed boundary value problems by conformal transformation methods. Z. angew. Math. Phys. 23, 655-664, 1972  8.    Papamichael, N. and Whiteman, J.R., A numerical conformal transformation method for harmonic mixed boundary value problems in polygonal domains. Z. angew. Math. Phys. 24, 304-316,1973.  9.    Thatcher, R.W., Singularities in the solution of Laplace's equation in two dimensions. J. Inst. Maths. Applies. 16, 303-319,1975.  10.   Wait, R. and Mitchell, A.R., Corner singularities in elliptic problems by finite element methods. J. Comp. Phys. 8, 45-52, 1971.  11.   Webb, J.C. and Whiteman, J.R., Convergence of finite-difference techniques for a harmonic mixed boundary value problem. B.I.T. 10, 366-374, 1970.  12.   Whiteman. J.R., Singularities due to re-entrant corners in harmonic boundary-value problems.  Technical Report No.829, Mathematics Research Center, University of Wisconsin, Madison, 1967. - 15 –  13.   Whiteman, J.R., Treatment of singularities in a harmonic mixed boundary value problem by dual series methods.  Q.J. Mech. Appl. Math. 21, 41-50, 1968.  14.   Whiteman, J.R., Numerical solution of a harmonic mixed boundary value problem by the extension of  a  dual  ser ies  method.  Q.J .  Mech. Appl .  Math.  23, 449-455, 1970.   15.   Woods, L.C., The relaxation treatment of singular points in Poisson 's  equa t ion .  Q.J .  Mech.  Appl .  Math .  6 ,  163-189, 1953.  

A method of treating boundary singularities in time-dependent problems.

A numerical conformal transformation method for harmonic mixed boundary value problems in polygonal domains.

A power series solution for a harmonic mixed boundary value problem.

A simple finite-difference modification for improving accuracy near a corner in heat flow problems.

Convergence of finite-difference techniques for a harmonic mixed boundary value problem.

Corner singularities in elliptic problems by finite element methods.

Local mesh refinement with finite elements for elliptic problems.

Numerical solution of a harmonic mixed boundary value problem by the extension of a dual series method.

Singularities due to re-entrant corners in harmonic boundary-value problems.

Singularities in the solution of Laplace's equation in two dimensions.

The finite element method for non-smooth domains and coefficients,

The relaxation treatment of singular points in Poisson's equation.

The treatment of singularities of partial differential equations by relaxation methods.

Treatment of harmonic mixed boundary value problems by conformal transformation methods.

Treatment of singularities in a harmonic mixed boundary value problem by dual series methods.

Modified finite-difference approximations near the singularitiy in the problem of motz

J. Crank

R. M. Furzel

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MODIFIED FINITE-DIFFERENCE APPROXIMATIONS NEAR THE SINGULARITY IN THE PROBLEM OF MOTZ

http://bura.brunel.ac.uk/bitstream/2438/1969/1/TR_60.pdf

Modified finite-difference approximations near the singularitiy in the problem of motz

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