Development of a modified Runge-Kutta scheme with TVD limiters for ideal three-dimensional magnetogasdynamics

a b s t r a c tAn E-CUSP (energy-convective upwind and split pressure) scheme is developed to solve the equations of magnetohydrodynamics. Fifth order WENO reconstructions are employed to calculate the fluxes in order to achieve high order spacial accuracy. A characteristic speed of sound by averaging the fast wave speed and the acoustic speed of sound is suggested to evaluate the Mach number, which will yield robust and accurate solutions. The numerical experiments have demonstrated the accuracy and the capability of the new scheme to capture complex interactions of multiple shocks and vortices.

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“…The ideal MHD equations for inviscid flow can be expressed in vector form as [39] @U @t þ r Á F ¼ 0;…”

Section: Governing Equationsmentioning

confidence: 99%

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

Shen

Zha

Huerta

2012

Journal of Computational Physics

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“…[13] introduced a series of eigenvectors for these eigenvalues. By looking at these eigenvalues and corresponding eigenvectors, it is clear that v sζ degenerates to the value of zero if B ξ equals zero.…”

Section: Eigensystemmentioning

confidence: 99%

“…Due to the mathematical properties of this hyperbolic system of the ideal eight-wave MHD equations mentioned earlier, the introduction of TVD scheme to solve this system has attracted extensive attentions [10][11][12][13]. However, it is difficult since it requires the determination of the eigenvalues and eigenvectors for the system of equations, which has been proved to be not strictly with non-genuine nonlinearity (non-convexity) of some of the local wave fields.…”

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confidence: 97%

Simulation of three-dimensional nonideal MHD flow at high magnetic Reynolds number

Hao

2010

Sci. China Ser. E-Technol. Sci.

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A conservative TVD scheme is adopted to solve the equations governing the three-dimensional flow of a nonideal compressible conducting fluid in a magnetic field. The eight-wave equations for magnetohydrodynamics (MHD) are proved to be a non-strict hyperbolic system, therefore it is difficult to develop its eigenstructure. Powell developed a new set of equations which cannot be numerically simulated by conservative TVD scheme directly due to its non-conservative form. A conservative TVD scheme augmented with a new set of eigenvectors is proposed in the paper. To validate this scheme, 1-D MHD shock tube, unsteady MHD Rayleigh problem and steady MHD Hartmann problem for different flow conditions are simulated. The simulated results are in good agreement with the existing analytical results. So this scheme can be used to effectively simulate high-conductivity fluids such as cosmic MHD problem and hypersonic MHD flow over a blunt body, etc. magnetohydrodynamics (MHD), numerical simulation, TVD, MHD Rayleigh flow, MHD Hartmann flow Citation:Lu H Y, Lee C H. Simulation of three-dimensional nonideal MHD flow at highThe physical mechanisms in space and near-space physics are involved in the interaction of aerodynamics and electromagnetic dynamics. For instance, investigations have indicated that the strength of local magnetic field on the solar surface is of the order of kilogauss, and the elements of gas in the vicinity of solar surface are almost fully ionized. Therefore, the physical activities on solar surface are controlled by the interactions of strong magnetic field and plasma [1-3]. Renewed scientific interest has arisen all over the world for the potential application of MHD processes for the advancement of flight vehicles. In the early nineties of the last century, Russians proposed a new concept of combined propulsion system which was composed of a turbine and an MHD enhanced scramjet for hypersonic cruised aircraft. Later, they established the AJAX program [4]. The most innovative and key issue about this program was the utilization of MHD system to develop an MHD bypass scramjet technology, and to control the inlet of the propulsion system. According to the AJAX program, the inlet control can be furnished by an MHD device through the modification of the external flow upstream of the scramjet inlet. It helps in improving the performance of engine at off-design conditions [3][4][5][6]. The governing equations for MHD include the conservation laws of fluid dynamics augmented by terms including magnetic effects and Maxwell's equation of electrodynamics. It forms an eight-wave model demonstrating the strong interaction between the flow field and electromagnetic field. This system of equations is not strictly hyperbolic since eigenvalues of the system may coincide, causing the system to be nonconvex [7,8]. Due to these mathematical characteristics, the algorithms implemented on Euler equations do not guarantee the solution of the eight-wave equations.

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“…Hence the eigenstructure of MHD system is very complex. This results in the complexity of those numerical methods based on the eigenstructure [1][2][3][4][5][6]. The convective upwind and split pressure(CUSP) schemes, which consider the convective upwind characteristics of flow and do not require the eigen-decomposition process, have been extended to calculate the MHD equations recently [7,8].…”

Section: Introductionmentioning

confidence: 99%

“…Since the equations of ideal Magnetohydrodynamics (MHD) have a wave-like structure analogous to that of the hydrodynamics equations, various numerical schemes for hydrodynamics equations have been extended to solve the MHD equations [1][2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Non-Oscillatory Finite Compact Scheme for the Equations of Ideal Magnetohydrodynamics

Liu

Shen

2013

AMM

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Abstract. Although the equations of ideal Magnetohydrodynamics (MHD) is a non-strictly hyperbolic system, they have a wave-like structure analogous to that of the hydrodynamics equations, various numerical schemes for hydrodynamics equations have been extended to solve the MHD equations. The finite compact (FC) scheme treats the discontinuity as the internal boundary and avoids the global dependence of the traditional compact schemes. By using a parameter-free shock detecting method, the computational domain is divided into a series of smooth regions and shock wave regions. In the shock wave regions, the shock capturing scheme is used to construct the numerical flux, and in the smooth regions the compact scheme is used, the flux of shock wave region is automatically the boundary formulation of the compact scheme. Hence, the FC scheme can resolve shock essentially non-oscillatory and achieve high order of accuracy in smooth region. This paper develops the non-oscillation finite compact scheme for the ideal MHD equations.

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Development of a modified Runge-Kutta scheme with TVD limiters for ideal three-dimensional magnetogasdynamics

Cited by 6 publications

References 9 publications

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

Simulation of three-dimensional nonideal MHD flow at high magnetic Reynolds number

Non-Oscillatory Finite Compact Scheme for the Equations of Ideal Magnetohydrodynamics

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