$$u_t + \left[f(u)\right]_x = 0$$

# Finite Volume

$$w(x, t) = \frac{1}{\Delta x} \int_{x - \Delta x / 2}^{x + \Delta x / 2} u(s, t) \, ds$$
\begin{align*} w_t &= \frac{1}{\Delta x} \int_{x - \Delta x / 2}^{x + \Delta x / 2} u_t \, ds \\ &= \frac{1}{\Delta x} \int_{x - \Delta x / 2}^{x + \Delta x / 2} -\left[f(u)\right]_s \, ds \\ &= -\frac{f\left(u\left(x + \Delta x / 2, t\right)\right) - f\left(u\left(x - \Delta x / 2, t\right)\right)}{\Delta x} \end{align*}
\begin{align*} u_j^n &= u\left(j \Delta x, n \Delta t\right) \\ w_j^n &= w\left(j \Delta x, n \Delta t\right) \end{align*}
$$\frac{d}{dt} w_j^n = -\frac{1}{\Delta x}\left(f\left(u_{j + \frac{1}{2}}\right) - f\left(u_{j - \frac{1}{2}}\right)\right)$$
$$\left\{w_j^n\right\}_j \overset{?}{\longrightarrow} \left\{u_{j + \frac{1}{2}}^n\right\}_j$$

WENO interpolation ... [is] used ... to transfer information from one domain to another in a high order, nonoscillatory fashion

# Finite Difference

$$\frac{d}{dt} u_j^n = -\frac{1}{\Delta x}\left(\widehat{f}_{j + \frac{1}{2}} - \widehat{f}_{j - \frac{1}{2}}\right)$$

Define $h(x)$ implicitly via $$\frac{1}{\Delta x} \int_{x - \Delta x / 2}^{x + \Delta x / 2} h(s) \, ds = f(u(x))$$

$$\frac{1}{\Delta x} \int_{x - \Delta x / 2}^{x + \Delta x / 2} h(s) \, ds = f(u(x))$$
$$\Longrightarrow \frac{1}{\Delta x} \left(h\left(x_{j + \frac{1}{2}}\right) -h\left(x_{j - \frac{1}{2}}\right)\right) = \left[f(u)\right]_x$$
$$\Longrightarrow \frac{1}{\Delta x} \left(h\left(x_{j + \frac{1}{2}}\right) -h\left(x_{j - \frac{1}{2}}\right)\right) \approx \frac{1}{\Delta x}\left(\widehat{f}_{j + \frac{1}{2}} - \widehat{f}_{j - \frac{1}{2}}\right)$$
$$\widehat{f}_{j + \frac{1}{2}} = h\left(x_{j + \frac{1}{2}}\right) = h_{j + \frac{1}{2}}$$
$$\overline{h}_j^n = \frac{1}{\Delta x} \int_{x - \Delta x / 2}^{x + \Delta x / 2} h(s) \, ds = f\left(u_j^n\right)$$
$$\left\{\overline{h}_j^n\right\}_j \overset{?}{\longrightarrow} \left\{h_{j + \frac{1}{2}}^n\right\}_j$$
In [3]:
weno_computations.make_intro_plots(0)

In [4]:
weno_computations.make_intro_plots(1)

In [5]:
weno_computations.make_intro_plots(2)

In [6]:
weno_computations.make_intro_plots(3)

In [7]:
weno_computations.make_intro_plots(4)

In [8]:
weno_computations.make_intro_plots(5)

In [9]:
weno_computations.make_intro_plots(6)

In [10]:
weno_computations.make_intro_plots(7)

In [11]:
weno_computations.make_intro_plots(8)

In [12]:
weno_computations.make_intro_plots(9)

In [13]:
weno_computations.make_intro_plots(10)

In [14]:
weno_computations.make_intro_plots(11)

In [15]:
weno_computations.make_intro_plots(12)

$$u^{{(1)}}_{{j + \frac{1}{2}}} = \frac{1}{3}\overline{u}_{j-2} - \frac{7}{6}\overline{u}_{j-1} + \frac{11}{6}\overline{u}_{j}$$
$$u^{{(2)}}_{{j + \frac{1}{2}}} = -\frac{1}{6}\overline{u}_{j-1} + \frac{5}{6}\overline{u}_{j} + \frac{1}{3}\overline{u}_{j+1}$$
$$u^{{(3)}}_{{j + \frac{1}{2}}} = \frac{1}{3}\overline{u}_{j} + \frac{5}{6}\overline{u}_{j+1} - \frac{1}{6}\overline{u}_{j+2}$$
$$u^{{(\ast)}}_{{j + \frac{1}{2}}} = \frac{1}{30}\overline{u}_{j-2} - \frac{13}{60}\overline{u}_{j-1} + \frac{47}{60}\overline{u}_{j} + \frac{9}{20}\overline{u}_{j+1} -\frac{1}{20}\overline{u}_{j+2}$$
$$\frac{1}{3}\overline{u}_{j-2} - \frac{7}{6}\overline{u}_{j-1} + \frac{11}{6}\overline{u}_{j} = u\left(x_{j + \frac{1}{2}}\right) - \frac{\Delta x^3 u_0'''}{4} + \mathcal{O}\left(\Delta x^4\right)$$
$$-\frac{1}{6}\overline{u}_{j-1} + \frac{5}{6}\overline{u}_{j} + \frac{1}{3}\overline{u}_{j+1} = u\left(x_{j + \frac{1}{2}}\right) + \frac{\Delta x^3 u_0'''}{12} + \mathcal{O}\left(\Delta x^4\right)$$
$$\frac{1}{3}\overline{u}_{j} + \frac{5}{6}\overline{u}_{j+1} - \frac{1}{6}\overline{u}_{j+2} = u\left(x_{j + \frac{1}{2}}\right) - \frac{\Delta x^3 u_0'''}{12} + \mathcal{O}\left(\Delta x^4\right)$$
$$u^{{(\ast)}}_{{j + \frac{1}{2}}} = u\left(x_{j + \frac{1}{2}}\right) - \frac{\Delta x^5 u_0'''}{60} + \mathcal{O}\left(\Delta x^6\right)$$
$$\left(\Delta x\right) \overline{u} = \int_a^b u(x) \, dx = u_0(b - a) + u_0' \left(\frac{b^2 - a^2}{2}\right) + u_0'' \left(\frac{b^3 - a^3}{6}\right) + \cdots$$

Putting the W in WENO: $$u^{{(\ast)}}_{{j + \frac{1}{2}}} = \frac{1}{10} u^{{(1)}}_{{j + \frac{1}{2}}} + \frac{6}{10} u^{{(2)}}_{{j + \frac{1}{2}}} + \frac{3}{10} u^{{(3)}}_{{j + \frac{1}{2}}}$$

What about the other letters?

What about shocks?

In [17]:
weno_computations.discontinuity_to_volume_single_cell(0)

In [18]:
weno_computations.discontinuity_to_volume_single_cell(1)

In [19]:
weno_computations.make_shock_plot_single_cell()

$$\frac{1}{10} u^{{(1)}}_{{j + \frac{1}{2}}} + \frac{6}{10} u^{{(2)}}_{{j + \frac{1}{2}}} + \frac{3}{10} u^{{(3)}}_{{j + \frac{1}{2}}} \longrightarrow \frac{2}{3} u^{{(2)}}_{{j + \frac{1}{2}}} + \frac{1}{3} u^{{(3)}}_{{j + \frac{1}{2}}}$$
In [20]:
weno_computations.discontinuity_to_volume()

In [21]:
weno_computations.make_shock_plot()


Smoothness indicator when approximating a point in the interval $I = \left(x_j - \frac{\Delta x}{2}, x_j + \frac{\Delta x}{2}\right]$

$$\beta = \sum_k \Delta x^{2k - 1} \int_I \left(\frac{d^k}{dx^k} p\right)^2 dx$$

Large $\beta$ indicates that the approximation $p(x)$ is not smooth nearby our point.

Recall the W in WENO: $$u^{{(\ast)}}_{{j + \frac{1}{2}}} = \frac{1}{10} u^{{(1)}}_{{j + \frac{1}{2}}} + \frac{6}{10} u^{{(2)}}_{{j + \frac{1}{2}}} + \frac{3}{10} u^{{(3)}}_{{j + \frac{1}{2}}}$$

Rather than using these directly, incorporate a penalty for each polynomial approximation's (lack of) smoothness: $\beta^{(1)}$, $\beta^{(2)}$, $\beta^{(3)}$.

$$\widetilde{w}^{(1)} = \frac{0.1}{\left(\varepsilon + \beta^{(1)}\right)^2}, \; \widetilde{w}^{(2)} = \frac{0.6}{\left(\varepsilon + \beta^{(2)}\right)^2}, \; \widetilde{w}^{(3)} = \frac{0.3}{\left(\varepsilon + \beta^{(3)}\right)^2}$$
$$w^{(j)} = \frac{\widetilde{w}^{(j)}}{\widetilde{w}^{(1)} + \widetilde{w}^{(2)} + \widetilde{w}^{(3)}}$$
$$w^{(1)} u^{{(1)}}_{{j + \frac{1}{2}}} + w^{(2)} u^{{(2)}}_{{j + \frac{1}{2}}} + w^{(3)} u^{{(3)}}_{{j + \frac{1}{2}}}$$

# Back to Conservation Law

$$\dot{\overline{u}} = -\frac{1}{\Delta x}\left(\widehat{f}\left(u_{j + \frac{1}{2}}^-, u_{j + \frac{1}{2}}^+\right) - \widehat{f}\left(u_{j - \frac{1}{2}}^-, u_{j - \frac{1}{2}}^+\right)\right)$$
$$u_{j + \frac{1}{2}}^- \longrightarrow \text{Reconstruct from } \overline{u}_{j - 2}, \overline{u}_{j - 1}, \overline{u}_{j}, \overline{u}_{j + 1}, \overline{u}_{j + 2}$$
$$u_{j + \frac{1}{2}}^+ \longrightarrow \text{Reconstruct from } \overline{u}_{j - 1}, \overline{u}_{j}, \overline{u}_{j + 1}, \overline{u}_{j + 2}, \overline{u}_{j + 3}$$
$$\dot{\overline{u}} = L\left(\overline{u}\right)$$

TV-diminishing / strong stability preserving RK scheme

$$\begin{array}{c | c c c} 0 & & & \\ 1 & 1 & & \\ 1/2 & 1/4 & 1/4 & \\ \hline & 1/6 & 1/6 & 2/3 \end{array}$$