Numerical Methods for Partial Differential Equations

This study material covers numerical methods for solving partial differential equations, including Triangular domain, Rothe's method, and 1D convention-diffusion-reaction. It includes detailed derivations and boundary conditions for each method. References are also provided for further reading.

3. Triangular domain
Given;
∆ u=f
Let Ω=Ω∩ R+¿2 ¿ amd Ω :=Ω ∪0< x , y <1
Taking i=0, we discretize as follows;
∂n
2 u
∂ x2 ( 0 ,1−x ) = 1
h2 ( u−1 , j−2 u0 , j +ui , j ) +O ( h2 ) … … … … …(i)
Applying the boundary conditions;
−γ0 , j= ∂n u
∂ x ( 0 , 1−x )= 1
2 h ( u1 , j −u−1 , j ) +O ( h2 ) … …… … …… .(ii)
Combining (i) and (ii);
∂n
2 u
∂ x2 ( t )= 1
h2 ( 2ui , j−2 u0 , j ) + 2 γ o , j
h +O(h2)
Where γ=0;
∂n
2 u
∂ x2 ( t )= 1
h2 ( 2ui , j−2 u0 , j ) +O( h2)
From the solution obtained we observe that the consistency level is 2.
From the solution we obtain that an alternating sequence is formed meaning that north point is
equal to the east point as well as west point being equal to the south point as described in the
figure below (Khasminskii, 2011).

From the details in the diagram we come up with a matrix as below;
(−4 1 0
1 −4 1
0 1 −4
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
1 0 0
0 1 0
0 0 1
−4 1 0
1 −4 1
0 1 −4
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
1 0 0
0 1 0
0 0 1
−4 1 0
1 −4 1
0 1 −4
)(u 1,1
u 2,1
u 3,1
u 1,2
u 2,2
u 3,2
u 1,3
u 2,3
u 3,3
)=
( h2 f 1,1−u 1,0−u 0,1
h2 f 2,1−u 2,0
h2 f 3,1−u 3,0−u 4,1
h2 f 1,2−u 0,2
h2 f 2,2
h2 f 3,2−u 4,2
h2 f 1,3−u 0,3−u1,4
h 62 f 2,3−u 2,4
h2 f 3,3−ui4,3−u 3,4
)4. Rothe’s method
uk +1=uk + τ ∆ uk+1 ,(x , y) Ω
Rearranging we have;
uk +1−uk =τ ∆ uk +1
Using the 5-point finite difference formula to discrete the PDE in space we obtain;
dui , j
dt ( x , y ) = D
h2 ( ui+1 , j +ui−1 , j+ui , j+1 +ui , j−1−4 ui , j ) +O(h2 )
Therefore,
uk+1 −uk
h =ui+1 , j
k+1 +ui−1 , j
k+ 1 −4 ui , j
k+1 +ui , j+ 1
k +1 +ui, j−1
k+1
h2 +O(h2 )
Multiplying both sides by h;
uk +1−uk =ui+1 , j
k+1 + ui−1 , j
k+1 −4 ui, j
k+1 +ui , j+1
k +1 +ui , j−1
k+1
h +O( h2)
The convergence order is 2.
To make the linear stable; τ ≤ h2
Comparing the 5 point finite difference formula to the implicit method, the implicit method is
inefficient.

5. 1D convention-diffusion-reaction
a)
For central difference formula; (Moin, 2010)
u' ' −b u'= u ( x+h )−2 u ( x )+u ( x−h )
h2 −b ( u ( x+ h )−u ( x−h )
2 h +O ( h2 ) )
Applying the boundary conditions;
u' ' −b u'= u ( x+ h ) −2 ( 1 ) +u ( x−h )
h2 −b ( u ( x +h ) −u ( x−h )
2 h + O ( h2 ))
¿ u ( x +h ) −2+u ( x−h )
h2 −b ( u ( x +h ) −u ( x−h )
2 h + O ( h2 ))
Forward difference formula;
u' ' −b u'= u ( x+ h ) −2 u ( x ) +u ( x−h )
h2 −b ( u ( x+ h ) −u ( x )
h +O (h))
Applying the boundary conditions;
u' ' −b u'= u ( x+h )−2 ( 1 ) +u ( x−h )
h2 −b (u ( x +h ) −1
h +O(h) )
u' '−b u'= u ( x+ h ) −2+u ( x−h )
h2 −b ( u ( x+ h ) −1
h + O(h))
Backward difference formula;
u' ' −b u' = u ( x+ h ) −2 u ( x ) +u (x−h)
h2 −b ( u ( x +h ) −u ( x−h)
h )
Applying the conditions;

u' ' −b u'= u ( x+ h )−2 ( 1 ) +u(x−h)
h2 −b ( u ( x +h )−u(x−h)
h )
u' ' −b u'= u ( x+h )−2+u ( x−h)
h2 −b ( u ( x+ h )−u (x−h)
h +O( h) )
b)
i) The solution for central difference is
u' ' −bu '=u ( x +h ) −2+u ( x −h )
h2 −b ( u ( x +h ) −u ( x −h )
2 h +O ( h2 ) )
To ensure its discrete maximum principle satisfied; h<0
ii) The solution for forward difference is
u' '−b u'= u ( x+ h ) −2+u ( x−h )
h2 −b ( u ( x+ h ) −1
h + O(h))
To ensure its discrete maximum principle satisfied; h<0
iii) The solution for backward difference is
u' ' −b u'= u ( x+h )−2+u ( x−h)
h2 −b ( u ( x+ h )−u (x−h)
h +O( h) )
To ensure its discrete maximum principle satisfied; h<0
i)

References
Khasminskii, R. (2011). Stochastic Stability of Differential Equations: In G. N. Milstein, Stochastic
Modelling and Applied Probability (vol 66). Springer Science & Business Media, pp 23-105.
Moin, P. (2010). Fundamentals of Engineering Numerical Analysis. Cambridge University Press, pp 10-33.