Lax-Wendroff and Godunov schemes for $u_t + (u^4)_x = 0$

hyperbolic-equationsnumerical methodspartial differential equations

Consider the nonlinear conservation law
$$ u_t + (u^4)_x = 0 $$
Write the following schemes for the equation in the form $u_j^{n+1} = F(u_{j-1}^{n},u_{j}^{n},u_{j+1}^{n})$.

Im trying to do for Lax-Wendroff and Godunov method.

Since this is advection, we know $u(x-a \Delta t, t^n) = u(x,t^{n+1})$. So I was thinking to interpolate the points $(x_{i}, u_i)$ at n-level for $i=j-1,j,j+1$ so as to obtain a quadratic and then using the above we may write a three-point stencil as asked. But, I don't really quite understand how to for Godunov. Any advice would be greatly appreciated.

Best Answer

The present conservation law $u_t + f(u)_x = 0$ is nonlinear, with a convex flux $f(u)=u^4$. The derivative of the flux is $f'(u)=4u^3$, and the only solution of $f'(u_s) = 0$ is $u_s=0$. The Lax-Wendroff method is well-described in the Wikipedia article. The method can be written in conservation form $$u_{j}^{n+1} = u_{j}^n - \frac{\Delta t}{\Delta x} (F_{j+1/2}^n - F_{j-1/2}^n) $$ with the numerical flux $$ F_{j+1/2}^n = \frac{1}{2} \left({f(u_j^n) + f(u_{j+1}^n)}\right) - \frac12 \frac{\Delta t}{\Delta x} A_{j+1/2} \left(f(u_{j+1}^n)-f(u_{j}^n)\right) $$ where $ A_{j+ 1/2}=f'\big(\tfrac12(u_{j}^n + u_{j+ 1}^n)\big) $, which is of the desired form. Godunov's method is usually written in conservation form too, with the numerical flux (see (1) p. 228) $$ F_{j+1/2}^n = \left\lbrace \begin{aligned} &f(u_j^n) & &\text{if}\quad u_j^n > u_s \;\text{and}\; s_{j+1/2} > 0 ,\\ &f(u_{j+1}^n) & &\text{if}\quad u_{j+1}^n < u_s\;\text{and}\; s_{j+1/2} < 0 ,\\ &f(u_s) & &\text{if}\quad u_{j}^n < u_s < u_{j+1}^n , \end{aligned}\right. $$ where $ s_{j+1/2} = [{f(u_{j+1}^n) - f(u_j^n)}]/[{u_{j+1}^n - u_{j}^n}] $, which is also of the desired form.

(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge university press, 2002. doi:10.1017/CBO9780511791253

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[Math] Lax-Wendroff method for linear advection – Stability analysis

Recall how the Lax-Wendroff method is obtained in the constant-speed case [1]:

a Taylor series in time is written: $$ u(x,t_{n+1}) = u(x,t_{n}) + \Delta t\, u_t(x,t_{n}) + \frac{1}{2}\Delta t^2\, u_{tt}(x,t_{n}) + \dots $$
the time derivatives are eliminated using the PDE: $u_t = -c u_x$ and $u_{tt} = c^2 u_{xx}$.
the spatial derivatives are replaced by central finite-difference approximations.

Thus, the following scheme is obtained: $$ u_j^{n+1} = u_j^n - c\Delta t \frac{D_x u_j^n}{2 \Delta x} + \frac{c^2\Delta t^2}{2} \frac{\delta_x^2 u_j^n}{\Delta x^2} \, . $$ This method can be adapted to the variable velocity case.

Let us analyze the stability of the Lax-Wendroff scheme above (constant-speed case). Assuming a perturbation of the form $u_j^{n} = \xi^n \text{e}^{\text i k x_j}$, one has \begin{aligned} \xi &= 1 - c \Delta t \frac{\text{e}^{\text i k \Delta x} - \text{e}^{-\text i k \Delta x}}{2 \Delta x} + \frac{c^2 \Delta t^2}{2} \frac{\text{e}^{\text i k \Delta x} - 2 + \text{e}^{-\text i k \Delta x}}{\Delta x^2} \\ &= 1 - \text{i} \kappa \sin (k\Delta x) + \kappa^2 \left(\cos (k\Delta x) - 1\right) . \end{aligned} where $\kappa = c\frac{\Delta t}{\Delta x}$ is the Courant number. Thus, the squared modulus of the amplification factor is \begin{aligned} |\xi|^2 &= (1 - \kappa^2 (1-\cos(k\Delta x)))^2 + \kappa^2 (1-\cos^2(k\Delta x))\\ &= 1 - 2\kappa^2 (1-\cos(k\Delta x)) + \kappa^2 (1-\cos^2(k\Delta x)) + \kappa^4 (1-\cos(k\Delta x))^2 \\ &= 1 - \kappa^2 (1-\kappa^2) (1-\cos(k\Delta x))^2 \\ &= 1 - 4\kappa^2 (1-\kappa^2) \sin^4\left(\tfrac{1}{2}k\Delta x\right) . \end{aligned} Finally, the Lax-Wendroff scheme is Neumann-stable provided that $|\xi|^2 \leq 1$, which implies that the CFL condition $|\kappa| \leq 1$ is satisfied.

[1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.

Implementing Lax-Wendroff scheme for advection in matlab

You transformed the method equation wrongly, there are multiple sign errors. It should be \begin{align} u_j^{n+1} &=u_j^n - \frac{p}2(u_{j+1}^n - u_{j-1}^n) + \frac{p^2}{2}(u_{j+1}^n - 2u_j^n + u_{j-1}^n) \\ &= u_j^n (1-p^2) -\frac{p-p^2}2 u_{j+1}^n +\frac{p+p^2}2 u_{j-1}^n \end{align}

Implementing this results in a correct looking dynamic

F = @(x) sin(pi*(x-1)).^2 .* (1<x).*(x<2);

n = 200;
x0 = 0; xn = 6; 
x = linspace(x0,xn,n+1);
dx = x(2)-x(1);
t = 0;
dt = 0.1*dx;
p = dt/dx;
v = F(x);

%%% Scheme iterations
k=1
while t<4
    v(2:n) = (1-p^2)*v(2:n)-0.5*(p-p^2)*v(3:n+1)+0.5*(p+p^2)*v(1:n-1);
    v(1) = v(2);
    v(n+1) = v(n);
    t = t + dt;
    if t>k
        subplot(4,1,k);
        hold on
        plot(x,v,'bo');
        plot(x,F(x-t),'k-');
        ylim([-0.1,1.1]);
        grid on;
        title(sprintf("t=%.3f",t))
        hold off
        k=k+1
    end
end

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[Math] Lax-Wendroff method for linear advection – Stability analysis

Implementing Lax-Wendroff scheme for advection in matlab

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