Limiter-free discontinuity-capturing scheme for compressible gas dynamics with reactive fronts

Abstract

This work proposes a new spatial reconstruction scheme in finite volume frameworks. Different from long-lasting reconstruction processes which employ high order polynomials enforced with some carefully designed limiting projections to seek stable solutions around discontinuities, the current discretized scheme employs THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive sharpness to solve both smooth and discontinuous solutions. Due to the essentially monotone and bounded properties of THINC function, difficulties to solve sharp discontinuous solutions and complexities associated with designing limiting projections can be prevented. A new simplified BVD (Boundary Variations Diminishing) algorithm, so-called adaptive THINC-BVD, is devised to reduce numerical dissipations through minimizing the total boundary variations for each cell. Verified through numerical tests, the present method is able to capture both smooth and discontinuous solutions in Euler equations for compressible gas dynamics with excellent solution quality competitive to other existing schemes. More profoundly, it provides an accurate and reliable solver for a class of reactive compressible gas flows with stiff source terms, such as the gaseous detonation waves, which are quite challenging to other high-resolution schemes. The stiff C-J detonation benchmark test reveals that the adaptive THINC-BVD scheme can accurately capture the reacting front of the gaseous detonation, while the WENO scheme with the same grid resolution generates unacceptable results. Owing also to its algorithmic simplicity, the proposed method can become as a practical and promising numerical solver for compressible gas dynamics, particularly for simulations involving strong discontinuities and reacting fronts with stiff source term.

A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm

“…Since the value given by hyperbolic tangent function tanh(x) lays in the region of [−1, 1], the value of THINC [24], the effect of the sharpness parameter β on numerical dissipation of the THINC scheme has been investigated with approximate dispersion relation (ADR) analysis. It is concluded that (i) with β = 1.1, denoted by β s hereafter, THINC has much smaller numerical dissipation than TVD scheme with Minmod limiter [28], and has similar but slightly better performance than the Van Leer limiter [3]; (ii) with a larger β, denoted by β l , compressive or anti-diffusion effect will be introduced, which is preferred for discontinuous solutions.…”

Constructing higher order discontinuity-capturing schemes with upwind-biased interpolations and boundary variation diminishing algorithm

“…Following [28], in this work we use THINC functions with β of m-level to represent different steepness, to realize non-oscillatory and less-dissipative reconstructions adaptively for various flow structures. A THINC reconstruction functionq T k i (x) with β k gives the reconstructed values q L,T k i+1/2 and q R,T k i−1/2 , (k = 1, 2, .…”

“…In this study, we propose and test fifth order scheme with P 4 T 2 − BVD, seventh order scheme with P 6 T 3 − BVD, ninth order scheme with P 8 T 3 − BVD and eleventh order scheme with P 10 T 3 − BVD. According to previous study in [28], in all tests of the present study we use β 1 = 1.1 and β 2 = 1.8 for P 4 is devised to reduce numerical dissipation to capture sharp discontinuities. In the final stage, the parameter β can be chosen from 1.6 to 2.2.…”

“…The switch between WENO and THINC is guided by the BVD (Boundary Variation Diminishing) algorithms, which select reconstruction functions by minimizing the jumps of the reconstructed values at the cell boundaries. The spectral analysis and numerical experiments [28] reveal that THINC function with properly chosen steepness can outperform most existing MUSCL schemes for spatial reconstruction. A new scheme, the adaptive THINC-BVD scheme, was then designed under the BVD principle by employing THINC functions with different sharpnesses to solve both smooth and discontinuous solutions [28].…”

“…We have recently developed a new class of shock-capturing schemes [25,28,26,27,29] which hybrid polynomial based nonlinear reconstructions and the THINC (Tangent of Hyperbola for INterface Capturing) function as the reconstruction candidates. Realizing that polynomial-based reconstruction may not be suitable for discontinuities, in [25] a non-polynomial jump-like THINC function is employed to represent the discontinuity.…”