Motivated by the enlightenment that diminishing the jump at the cell boundary can effectively reduce numerical dissipation near the critical region, and a novel constrained interpolation profile conservative semi-Lagrangian method is proposed based on a newly designed boundary variation diminishing algorithm. First, a constrained interpolation profile conservative semi-Lagrangian scheme with the piecewise tangent of hyperbola for interface capturing scheme is proposed as one candidate to represent jump-like discontinuities. Second, the constrained interpolation profile conservative semi-Lagrangian scheme with a fourth-order weighted essentially non-oscillatory limiter is used as another candidate to keep the high-order and non-oscillatory reconstruction for smooth solutions. The selection criterion of these two candidates is designed by minimizing the total variations of the first derivative at cell boundaries. A unified pressure-based projection formulation with a fractional step procedure is implemented with the proposed scheme to simulate both compressible and incompressible flows. A variety of numerical tests are studied, including linear and nonlinear scalar wave transport problems and compressible and incompressible flow problems. Results show that the proposed method can effectively eliminate numerical oscillation and diffusion, suggesting it has great potential to be applied to various types of engineering problems including both compressible and incompressible flows.
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