Exploring Finite Difference Methods for Numerical PDE Solutions

This assignment delves into the numerical solution of partial differential equations (PDEs) using finite difference methods. It begins by discretizing a triangular domain and applying boundary conditions to derive a finite difference approximation, analyzing its consistency level. The assignment then explores Rothe's method for time discretization, employing a 5-point finite difference formula for spatial discretization and assessing the convergence order and stability conditions. Finally, it examines the 1D convection-diffusion-reaction equation, applying central, forward, and backward difference formulas, deriving the discrete maximum principle for each scheme. The document concludes with references to relevant literature on stochastic stability and numerical analysis. Desklib offers a wealth of similar solved assignments and past papers to aid students in their studies.

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.