2. Conservative Form¶
2.1. Conversion from Non-Conservative Form to Conservative Form¶
- Scalar conservation laws have the following Conservative Form in 1D (where \(F\) = Flux):
\[{{\partial u} \over {\partial t}} + {\partial \over {\partial x}} \left[ F(u(x,t)) \right] = 0\]
- Scalar conservation laws have the following Non-Conservative Form in 1D (where \(A\) = Jacobian):
\[{{\partial u} \over {\partial t}} + A(u(x,t)) {{\partial u} \over {\partial x}}= 0\]
- To find the Conservative Form from the Non-Conservative Form, we find the Flux, by equating the appropriate terms from the Conservative and Non-Conservative Forms:
\[{\partial \over {\partial x}} (F(u)) = A(u) {{\partial u} \over {\partial x}}\]
- LHS given by the definition of a function of a function:
\[{\partial \over {\partial x}} (F(u(x,t))) =
{{\partial F} \over {\partial u}} {{\partial u} \over {\partial x}} =
A(u) {{\partial u} \over {\partial x}}\]
- By comparing the above, we observe that:
\[{{\partial F} \over {\partial u}} = A(u)\]
- By integration:
\[F(u) = \int A(u) du\]
2.2. Conversion of the Burgers Equation from Non-Conservative to Conservative Form¶
- Non-Conservative Form:
\[{\partial u \over \partial t} + u {\partial u \over \partial x} = 0\]
- Identification of A(u):
\[A(u) = u\]
- Integration of A(u) to determine the flux F(u)
\[F(u) = \int A(u) du = \int u du = {{u^2} \over 2}\]
- Conservative Form:
\[{\partial u \over \partial t} + {\partial \over {\partial x}}{ \left( {u^2 \over 2} \right)} = 0\]
Conversion from Non-Conservative Form to Conservative Form could result in multiple Conservative Forms e.g.
\[u {\partial \over {\partial t}} + u^2 {\partial \over {\partial x}} = 0\]
Converts to
\[{\partial \over {\partial t}} \left( u^2 \over 2 \right) +
{\partial \over {\partial x}}{ \left( {u^3 \over 3} \right)} = 0\]
Is this now an alternate solution?
2.3. What is the Difference Between the Conservative and Non-Conservative Forms?¶
Using BD for the Flux term for both Conservative and Non-Conservative Forms (we had to do this as it doesn’t make sense to integrate the Non-Conservative Form):
2.3.1. Conservative Form¶
\[\sum {\partial \over {\partial x}} \left( {{u^2} \over 2} \right) =
{{u_1^2-u_0^2} \over {2 \Delta x}} +
{{u_2^2-u_1^2} \over {2 \Delta x}} +
{{u_3^2-u_2^2} \over {2 \Delta x}} =
{{u_3^2-u_0^2} \over {2 \Delta x}}\]
i.e. Conservative Form gives \(\text{flux}_{out} - \text{flux}_{in}\) which is how it should be
In other words, if:
\[{\partial u \over \partial t} + {\partial \over \partial x} (F(u)) = 0\]
Then the time derivative of the integral of the conserved quantity equals the difference between the outlet and inlet fluxes.
\[{d \over dt} \int_{out}^{in} u(x,t) dx = F(u(out,t))-F(u(in,t))\]
2.3.2. Non-Conservative Form¶
\[\sum u {{\partial u} \over {\partial x}}=
u_1 {(u_1-u_0) \over {\Delta x}} +
u_2 {(u_2-u_1) \over {\Delta x}} +
u_3 {(u_3-u_2) \over {\Delta x}} =
{{u_1 (u_1-u_0) + u_2 (u_2-u_1) + u_3 (u_3-u_2)} \over {\Delta x}}\]
i.e. Non-Conservative Form gives a growing function that is not conserving the flux! These are internal numerical sources that do not cancel out like in the conservative form.