Burgers equation

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Problem definition

$\frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}=\mu \frac{\partial^2 u}{\partial x^2}$

Domain

$x \in \left[-5,10\right]$

Initial Condition

$u(x,0) = \begin{cases} 0 & x \le 0 \\ 1 & x > 0 \end{cases}$

Boundary condition

$u(0,t)=0$

Exact solution

$u(x,t) = \begin{cases} 0 & x \le 0 \\ x/t & 0 < x < t \\ 1 & \mbox{otherwise} \end{cases}$

Numerical method

Space

Explicit Scheme (DRP)

${(\frac{\partial u}{\partial x})}_i=\frac{1.0}{dx}\sum_{k=-3}^3 a_k u_{i+k}$

The coefficients can be found in Tam(1993).At the right boundaries use fourth order central difference and fourth backward difference.At left boundaries use second order central difference for i=2 and fourth order central difference for i=3.The Dispersion relation preserving (DRP) finite volume scheme can be found in Popescu (2005).

Implicit Scheme(Compact)

Domain: $\alpha v_{i-1} + v_i + \alpha v_{i+1}=\frac{a}{2h}(u_{i+1}-u_{i-1})$

Boundaries: $v_1+\alpha v_2=\frac{1}{h}(au_1+bu_2+cu_3+du_4)$

where v refers to the first derivative.For a general treatment of compact scheme refer to Lele (1992).In this test case the following values are used

$\mbox{Domain:} \alpha=0.25 , a=\frac{2}{3}(\alpha+2)$

$\mbox{Boundary:} \alpha=2 ,a=-(\frac{11+2\alpha}{6}),b=\frac{6-\alpha}{2},c=\frac{2\alpha-3}{2},d=\frac{2-\alpha}{6}$

Both the schemes are 4th order accurate in the domain.The compact scheme has third order accuracy at the boundary.

Time (4th Order Runga-Kutta)

$\frac{\partial u}{\partial t}=f$

$u^{M+1} =u^M + b^{M+1}dtH^M$

$H^M=a^MH^{M-1}+f^M$

,M=1,2..5 .The coefficients a and b can be found in Williamson(1980)

Results

Reference

Mihaela Popescu, Wei Shyy , Marc Garbey (2005), "Finite volume treatment of dispersion-relation-preserving and optimized prefactored compact schemes for wave propagation", Journal of Computational Physics, Vol. 210, pp. 705-729.

Tam and Webb (1993), "Dispersion-relation-preserving finite difference schemes for computational acoustics", Journal of Computational Physics, Vol. 107, pp. 262-281.

SK Lele (1992), "Compact finite difference schemes with spectrum-like resolution", Journal of Computational Physics, Vol.103, pp.16-42.

Williamson (1980), "Low Storage Runge-Kutta Schemes", Journal of Computational Physics, Vol.35, pp.48–56.