A partition of unity finite element method for computational diffusion MRI
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It is critical for a numerical scheme to obtain numerical results as accurate as possible with limited computational resources. Turbulent processes are very sensitive to numerical dissipation, which may dissipate the small length scales. On the other hand, dealing with shock waves, capturing and reproducing of the discontinuity may lead to non-physical oscillations for non-dissipative high-order schemes. In the present work, a new high-order mixed weighted compact and non-compact difference scheme (MWCS hereafter) is proposed for accurate approximation of the derivatives in the governing Euler equations. The basic idea is to recover the non-dissipative high-order weighted compact scheme (WCS) in smooth regions, while linearly combine the WCS with a non-compact scheme, the weighted essentially non-oscillatory (WENO) scheme, for near-shock areas, by using a shock-detecting function. The proposed formulation does not involve any case-dependent adjustable parameter. A detailed Fourier and local truncation error analysis are used for assessing the dispersion and dissipation characteristics of the scheme. Numerical tests are performed for the one- and two-dimensional case and the results are compared with the well-established WENO scheme and the WCS.