3.3.1 Weak solutions

Conservation law:

$$\frac{\partial {\bf u}}{\partial t} + \frac{\partial {\bf f}({\bf u})}{\partial x} = 0$$ (89)

Test function: $\Phi(x,t) \in C_0^1(R\times R)$

$C_0^1$ is the space of continuously differentiable functions with compact support

Multiplying the conservation law by $\Phi$, integrating over space and time and integrating by parts, we get

$$\int_0^{\infty}\int_{-\infty}^{+\infty} (\Phi_t {\bf u} + \P... ...})) dx dt = -\int_{-\infty}^{+\infty} \Phi(x,0){\bf u}(x,0) dx$$ (90)

Definition: The function ${\bf u}(x,t)$ is called a weak solution of the conservation law if the previous integral form holds for all functions $\Phi\in C_0^1(R\times R)$

Theorem: Let ${\bf u}$ be a piecewise smooth function. Then, ${\bf u}$ is a solution of the integral form of the conservation system if and only if the two following conditions are satisfied:

1. ${\bf u}$ is a classical solution in the domains where it is continuous.
2. Across a given surface of discontinuity, $\Sigma$, it satisfies the jump conditions (Rankine-Hugoniot conditions)
   

   $$({\bf u}_R - {\bf u}_L)n_t + \sum_{j=1}^d\left[{\bf f}_j({\bf u}_R) - {\bf f}_j({\bf u}_L)\right]n_{xj} = 0,$$ (91)

   
   
   where ${\bf u}_L$ and ${\bf u}_R$ stand, respectively, for the values of ${\bf u}$ on the left and right hand sides of $\Sigma$, and ${\bf n}=(n_t,n_{x1},n_{x2},\ldots,n_{xd})$ denotes a vector normal to $\Sigma$.

For 1D systems, the Rankine-Hugoniot jump condition (92) reduces to

$$s({\bf u}_R - {\bf u}_L) = {\bf f}({\bf u}_R) - {\bf f}({\bf u}_L)$$ (92)

where $s$ is the propagation velocity of the discontinuity.

Rankine-Hugoniot conditions follow from the conservation of fluxes across the surfaces of discontinuity.

Shock-tracking techniques: use the RH conditions in combination with standard finite-difference methods for the smooth regions and special procedures for tracking the location of discontinuities $\rightarrow$ solving the equations in the presence of shocks

In 1D this is often a viable approach.

In multidimensional applications it is more cumbersome: the discontinuities lie along curves (in 2D) or surfaces (in 3D) and in realistic problems there may be many such discontinuities interacting in complicated ways.