Very-high-order weno schemes

Numerical Methods in Fluids

et al. 2019

Summary In this paper, we propose a parameter‐free algorithm to calculate ε, a parameter of small quantity initially introduced into the nonlinear weights of weighted essentially nonoscillatory (WENO) scheme to avoid denominator becoming zero. The new algorithm, based on local smoothness indicators of fifth‐order weighted compact nonlinear scheme (WCNS), is designed in a manner to adaptively increase ε in smooth areas to reduce numerical dissipation and obtain high‐order accuracy, and decrease ε in discontinuous areas to increase numerical dissipation and suppress spurious numerical oscillations. We discuss the relation between critical points and discontinuities and illustrate that, when large gradient areas caused by high‐order critical points are not well resolved with sufficiently small grid spacing, numerical oscillations arise. The new algorithm treats high‐order critical points as discontinuities to suppress numerical oscillations. Canonical numerical tests are carried out, and computational results indicate that the new adaptive algorithm can help improve resolution of small scale flow structures, suppress numerical oscillations near discontinuities, and lessen susceptibility to flux functions and interpolation variables for fifth‐order WCNS. The new adaptive algorithm can be conveniently generalized to WENO/WCNS with different orders.

\begin{matrix} alignleft \\ align-1 S O : {[ρ, u, p]}^{T} & align-2 = \{\begin{matrix} {false[3.857143, 2.629369, 10.333333 false]}^{T}, & x \in false[- 5, - 4 false] \\ {false[1 + 0.2 \sin 5 x, 0, 1 false]}^{T}, & x \in false(- 4, 5 false] \end{matrix} \\ align-1 T T : {[ρ, u, p]}^{T} & align-2 = \{\begin{matrix} {false[1.515695, 0.523346, 1.805 false]}^{T}, & x \in false[- 5, - 1.5 false] \\ {false[1 + 0.1 \sin 20 π x, 0, 1 false]}^{T}, & x \in false(- 1.5, 5 false] \end{matrix} \\ align-1 M T W W : {[ρ, u, p]}^{T} & align-2 = \{\begin{matrix} {false[0.6357, 0.4142, 1.4018 false]}^{T}, & x \in false[- 7, - 4 false] \\ {false[0.5 + 0.1 \sin 5 x, 0, 1 false]}^{T}, & x x \in false(- 4, 3 false], \end{matrix} \end{matrix}

with t end =1.8, 4, and 2 respectively, and CFL = 0...…”

Section: Numerical Testsmentioning

confidence: 99%

A parameter‐free ε‐adaptive algorithm for improving weighted compact nonlinear schemes

Zheng

Deng

Numerical Methods in Fluids

et al. 2019

“…It is well known that Roe's flux function produces a spurious triple point (also known as the kinked Mach stem) in this problem. The setups of the initial and boundary conditions are based on Ref . As shown in Figure , the shock wave, inclined at 60° with respect to the x ‐axis and propagated to the right at M a = 10, against a region of still air (state

{([], ρ, u, v, p)}_{B}^{T} = {([], 1.4, 0, 0, 1)}^{T}

).…”

Section: Numerical Testsmentioning

confidence: 99%

“…The boundary conditions correspond to the exact shockwave motion at the upper boundary ( y = 1). More details can be found in . The computational domain is [0,4] × [0,1], and the mesh size is chosen to be 800 × 200.…”

Section: Numerical Testsmentioning

confidence: 99%

Developing a hybrid flux function suitable for hypersonic flow simulation with high‐order methods

Deng

Numerical Methods in Fluids

et al. 2015

SUMMARYIn this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5% extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high-order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high-order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock-capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high-order weighted compact nonlinear scheme is not only robust for shock-capturing but also accurate for hypersonic heat transfer prediction.

AIP Conference Proceedings

“…On each time step, we first compute predictorũ n+1 by using (7). Then, the Crank-Nicolson scheme is used to find the corrector u n+1 = u n + τ 2 W (u n )F(u n ) + G(t n ) +W (ũ n+1 )F(u n+1 ) + G(t n+1 ) .…”

Section: Time Discretizationmentioning

confidence: 99%

“…High order of accuracy WENO methods have been constructed on uniform meshes [7]. A fifth order WENO method with non-uniform meshes has been obtained in [8] by using Lagrange polynomials.…”

Section: Introductionmentioning

confidence: 99%

On the numerical solution of hyperbolic equations with singular source terms

Turk¹,

Ashyraliyev²

2014

Articles you may be interested inOn the numerical solution of diffusion problem with singular source terms AIP Conf.Abstract. A numerical study for hyperbolic equations having singular source terms is presented. Singular in the sense that within the spatial domain the source is defined by a Dirac delta function. Solutions of such problems will have discontinuities which forms an obstacle for standard numerical methods. In this paper, a fifth order flux implicit WENO method with non-uniform meshes is studied for approximate solutions of hyperbolic equations having singular source terms. Numerical examples are provided.