How do we get the right constant?

[evinda]

Hello! (Wave)We consider the initial - boundary value problem

$$u_t(t,x)=a(t,x) u_{xx}(t,x)-c(t,x) u(t,x) \forall t \in [0,T_f], x \in [a,b] \\ u(0,x)=u_0(x) \forall x \in [a,b] \\ u(t,a)=0=u(t,b) \forall t \in [0,T_f]\\ a,c \in C^1, a(t,x)>0, c(t,x) \geq 0$$

$$\tau=\frac{T_f}{N_t}, t_n=n \tau, n=0,1, \dots, N_t$$

We consider the explicit Euler method.

The sheme is $u_i^{n+1}= \mu a_i^n u_{i+1}^n +(1- 2 \mu a_i^n - \tau c_i^n) u_i^n + \mu a_i^n u_{i-1}^n$

$u_i^n=u_0(x_i)$ for $n=0, \forall i=0, \dots, N_x+1$

$u_i^n=0$ for $i=0, i=N_x+1$ , $\forall n=0, \dots, N_t$

Let $u$ be the exact solution of the problem.
Suppose that $\mu a_i^n +\frac{\tau}{2} c_i^n \leq \frac{1}{2} \forall i,n$.

Then $\max_{1 \leq i \leq N_x} |u(t_n, x_i)-u_i^n| \leq T_f \left( \frac{\tau}{2} M_{tt}+\frac{h^2}{12} M_{xxxx} A \right) $, where $A=||a(t,x)||_{\infty}, M_{tt}=||u_{tt}(t,x)||_{\infty}, M_{xxxx}=||u_{xxxx}||_{\infty}$.

I want to show the above inequality.$e_i^n:=u(t_n, x_i)-u_i^n, e_i^0=0$

I have shown that $E_n \leq (2+2 \mu A) E_{n-1}+\tau \left( \frac{\tau}{2} M_{tt}+ A \frac{h^2}{12} M_{xxxx}\right)$, where $E_n=\max_{1 \leq i \leq N_x} |e_i^n|$.But then we get that $E_n \leq \sum_{i=0}^{n-1} (2+ 2 \mu A)^i \tau \left( \frac{\tau}{2} M_{tt}+ A \frac{h^2}{12} M_{xxxx}\right)= \frac{(2+ 2 \mu A)^n-1}{2 \mu A+1} \tau \left( \frac{\tau}{2} M_{tt}+ A \frac{h^2}{12} M_{xxxx}\right) $.But then we don't get the right constant of the inequality.
So do you think that my calculations are wrong? (Thinking)

 

[jvicens]

Hello!

First of all, let me say that your calculations are correct. However, I believe there may be a slight misunderstanding in the way you are interpreting the inequality. Let me explain.

The inequality you have derived is of the form $E_n \leq C \tau$, where $C$ is a constant that depends on the problem parameters and $E_n$ is the error at time $t_n$. This means that as we take smaller and smaller time steps (i.e. as $\tau \rightarrow 0$), the error will also decrease and approach 0. In other words, the error is converging to 0 as we refine the time step.

Now, in order to obtain a bound on the error over the entire time interval $[0,T_f]$, we need to sum up the individual errors at each time step. This is where the constant $C$ comes into play. The constant $C$ tells us how fast the error is decreasing as we refine the time step. In other words, it tells us how many time steps we need to take in order to achieve a certain level of accuracy.

So, when you say that you "don't get the right constant of the inequality", what you are actually saying is that the constant is not as small as you would like it to be. This is not necessarily an issue with your calculations, but rather a limitation of the method itself. The explicit Euler method is known to have a relatively large error constant, and this is why it is not always the most accurate method for solving initial-boundary value problems.

I hope this helps clarify things for you. Let me know if you have any further questions.