MUSCL Scheme - Kurganov and Tadmor Central Scheme | Kurganov Tadmor Central Scheme

Kurganov and Tadmor Central Scheme

A precursor to the Kurganov and Tadmor (KT) central scheme, (Kurganov and Tadmor, 2000), is the Nessyahu and Tadmor (NT) central scheme, (Nessyahu and Tadmor, 1990). It is a Riemann-solver-free, second-order, high-resolution scheme that uses MUSCL reconstruction. It is a fully discrete method that is straight forward to implement and can be used on scalar and vector problems, and can be viewed as a modification to the Lax-Friedrichs (LxF) scheme. The algorithm is based upon central differences with comparable performance to Riemann type solvers when used to obtain solutions for PDE's describing systems that exhibit high-gradient phenomena.

The KT scheme extends the NT scheme and has a smaller amount of numerical viscosity than the original NT scheme. It also has the added advantage that it can be implemented as either a fully discrete or semi-discrete scheme. Here we consider the semi-discrete scheme.

The calculation is shown below:

$F^*_{i-\frac{1}{2}} =\frac{1}{2} \left\{ \left - a_{i - \frac{1}{2} } \left \right\}.$

$F^*_{i+\frac{1}{2}} =\frac{1}{2} \left\{ \left - a_{i + \frac{1}{2} } \left \right\}.$

Where the local propagation speed, is the maximum absolute value of the eigenvalue of the Jacobian of over cells given by

$a_{i \pm \frac{1}{2} } \left( t \right) = \max \left[ \rho \left( \frac{\partial F \left( u_{i} \left( t \right) \right)}{\partial u} \right) , \rho \left( \frac{\partial F \left( u_{i \pm 1} \left( t \right) \right)}{\partial u} \right), \right]$

where represents the spectral radius of

Beyond these CFL related speeds, no characteristic information is required.

The above flux calculation is sometimes referred to as local Lax-Friedrichs flux or Rusanov flux (Lax, 1954; Rusanov, 1961; Toro, 1999; Kurganov and Tadmor, 2000; Leveque, 2002).

An example of the effectiveness of using a high resolution scheme is shown in the diagram opposite, which illustrates the 1D advective equation, with a step wave propagating to the right. The simulation was carried out on a mesh of 200 cells, using the Kurganov and Tadmor central scheme with Superbee limiter and used RK-4 for time integration. This simulation result contrasts extremely well against the above first-order upwind and second-order central difference results shown above. This scheme also provides good results when applied to sets of equations - see results below for this scheme applied to the Euler equations. However, care has to be taken in choosing an appropriate limiter because, for example, the Superbee limiter can cause unrealistic sharpening for some smooth waves.

The scheme can readily include diffusion terms, if they are present. For example, if the above 1D scalar problem is extended to include a diffusion term, we get

for which Kurganov and Tadmor propose the following central difference approximation,

$\frac{d u_i}{d t} = - \frac{1}{\Delta x_i} \left + \frac{1}{\Delta x_i} \left.$

Where,

$P_{i + \frac{1}{2}} = \frac{1}{2} \left[ Q \left( u_{i}, \frac{u_{i+1} - u_i}{\Delta x_i} \right) + Q \left( u_{i+1}, \frac{u_{i+1} - u_i}{\Delta x_i} \right) \right],$

$P_{i - \frac{1}{2}} = \frac{1}{2} \left[ Q \left( u_{i-1}, \frac{u_{i} - u_{i-1}}{\Delta x_{i-1}} \right) + Q \left( u_{i}, \frac{u_{i} - u_{i-1}}{\Delta x_{i-1}} \right). \right]$

Full details of the algorithm (full and semi-discrete versions) and its derivation can be found in the original paper (Kurganov and Tadmor, 2000), along with a number of 1D and 2D examples. Additional information is also available in the earlier related paper by Nessyahu and Tadmor (1990).

Note: This scheme was originally presented by Kurganov and Tadmor as a 2nd order scheme based upon linear extrapolation. A later paper (Kurganov and Levy, 2000) demonstrates that it can also form the basis of a third order scheme. A 1D advective example and an Euler equation example of their scheme, using parabolic reconstruction (3rd order), are shown in the parabolic reconstruction and Euler equation sections below.

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