MUSCL scheme

In the study of partial differential equations, the MUSCL scheme is a finite volume method that can provide highly accurate numerical solutions for a given system, even in cases where the solutions exhibit shocks, discontinuities, or large gradients.

In this paper he constructed the first high-order, total variation diminishing (TVD) scheme where he obtained second order spatial accuracy.

We will consider the fundamentals of the MUSCL scheme by considering the following simple first-order, scalar, 1D system, which is assumed to have a wave propagating in the positive direction, Where

An example of this effect is shown in the diagram opposite, which illustrates a 1D advective equation with a step wave propagating to the right.

The piecewise linear approximations are obtained from Thus, evaluating fluxes at the cell edges we get the following semi-discrete scheme where

are the piecewise approximate values of cell edge variables, i.e., Although the above second-order scheme provides greater accuracy for smooth solutions, it is not a total variation diminishing (TVD) scheme and introduces spurious oscillations into the solution where discontinuities or shocks are present.

MUSCL based numerical schemes extend the idea of using a linear piecewise approximation to each cell by using slope limited left and right extrapolated states.

This results in the following high resolution, TVD discretisation scheme, Which, alternatively, can be written in the more succinct form, The numerical fluxes

correspond to a nonlinear combination of first and second-order approximations to the continuous flux function.

has been chosen, such as the Kurganov and Tadmor scheme (see below), the solution can proceed using standard numerical integration techniques.

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 Rusanov flux (also called the local Lax-Friedrichs flux) supplemented with high order reconstructions.

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.

An example of the effectiveness of using a high resolution scheme is shown in the diagram opposite, which illustrates the 1D advective equation

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.

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, Where, 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.

A later paper (Kurganov and Levy, 2000) demonstrates that it can also form the basis of a third order scheme.

It is possible to extend the idea of linear-extrapolation to higher order reconstruction, and an example is shown in the diagram opposite.

However, for this case the left and right states are estimated by interpolation of a second-order, upwind biased, difference equation.

again represent scheme dependent functions (of the limited reconstructed cell edge variables).

Parabolic reconstruction is straight forward to implement and can be used with the Kurganov and Tadmor scheme in lieu of the linear extrapolation shown above.

This increase in spatial order has certain advantages over 2nd order schemes for smooth solutions, however, for shocks it is more dissipative - compare diagram opposite with above solution obtained using the KT algorithm with linear extrapolation and Superbee limiter.

Having obtained the limited extrapolated states, we then proceed to construct the edge fluxes using these values.

With the edge fluxes known, we can now construct the semi-discrete scheme, i.e., The solution can now proceed by integration using standard numerical techniques.

The diagram opposite shows a 2nd order solution to G A Sod's shock tube problem (Sod, 1978) using the above high resolution Kurganov and Tadmor Central Scheme (KT) with Linear Extrapolation and Ospre limiter.

This illustrates clearly demonstrates the effectiveness of the MUSCL approach to solving the Euler equations.

The simulation was carried out on a mesh of 200 cells using Matlab code (Wesseling, 2001), adapted to use the KT algorithm and Ospre limiter.

The following initial conditions (SI units) were used: The diagram opposite shows a 3rd order solution to G A Sod's shock tube problem (Sod, 1978) using the above high resolution Kurganov and Tadmor Central Scheme (KT) but with parabolic reconstruction and van Albada limiter.

The simulation was carried out on a mesh of 200 cells using Matlab code (Wesseling, 2001), adapted to use the KT algorithm with Parabolic Extrapolation and van Albada limiter.

Various other high resolution schemes have been developed that solve the Euler equations with good accuracy.

An open source implementation of the Kurganov and Tadmor central scheme can be found in the external links below.

1D advective equation , with step wave propagating to the right. Shows the analytical solution along with a simulation based upon a first order upwind spatial discretization scheme.
1D advective equation , with step wave propagating to the right. Shows the analytical solution along with a simulation based upon a second order, central difference spatial discretization scheme.
An example of MUSCL type left and right state linear-extrapolation.
1D advective equation , with step wave propagating to the right. Shows the analytical solution along with a simulation based upon the Kurganov and Tadmor central scheme with SuperBee limiter.
An example of MUSCL type state parabolic-reconstruction.
1D advective equation , with step wave propagating to the right. Shows the analytical solution along with a simulation based upon the Kurganov and Tadmor Central Scheme with parabolic reconstruction and van Albada limiter.
High resolution simulation of Euler equations based on G A Sod's 'Shock Tube' problem. Shows the analytical solutions along with simulated (2nd order) solutions based upon the Kuganov and Tadmor Central Scheme with Linear Extrapolation and Ospre limiter.
High resolution simulation of Euler equations based on G A Sod's 'Shock Tube' problem - SI units. Shows the analytical solutions along with simulated (3rd order) solutions based upon the Kurganov and Tadmor Central Scheme with parabolic reconstruction and van Albada limiter.