In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The averaged values on the sub-cells of current and face neighboring main cells are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume method incorporating with $\mathrm{P_NP_M}$ technique but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume method and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. The Fourier analysis is performed to study the properties of the one-dimensional schemes. A WENO limiter based on the secondary reconstruction is constructed.
This work was supported by the National Natural Science Foundation of China (Grant Nos. U1430235, and 11672160).
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Figure 1
Illustration for sub-cells of a main-cell.
Figure 2
Stencils (grey part) of the SCFV schemes. (a) S2T4P2 and S2T4P3 scheme; (b) S3T5P3 and S3T5P4 scheme.
Figure 3
Illustration for the flux point of a SCFV scheme.
Figure 4
Dispersion and dissipation properties of the third order S2T4P2 SCFV schemes, with upwind flux. (a) Dispersion relation; (b) relative dispersion error; (c) dissipation relation; (d) enlargement of dissipation error.
Figure 5
Dispersion and dissipation properties of fourth order S2T4P3 and S3T5P3 SCFV scheme, with upwind flux. (a) Dispersion relation; (b) relative dispersion error; (c) dissipation relation; (d) enlargement of dissipation error.
Figure 6
Dispersion and dissipation properties of the S2T4P2 type-2 scheme and the S3T5P3 type-2 scheme (do not recover the conservation on each sub-cell), uniformly partitioned, upwind flux. (a) Dispersion relation; (c) dissipation relation.
Figure 7
Dissipation properties of SV schemes and S3T5P3 schemes on meshes with non-uniform sub-cell partition, with upwind flux.
Figure 8
Thin lines in (a): contours for the magnitude of the amplification factor $G(Z)$ for the third order SSP Runge-Kutta method; thin lines in (b) and (c): contours for the magnitude of the amplification factor $G(Z)$ for the fourth order SSP Runge-Kutta method; thick line in (a): track of $Z = -I\sigma \kappa'$ for the S2T4P2 scheme with weight parameter $\gamma=3.0$ under the CFL number $\sigma = 0.573$; thick line in (b): track of $Z = -I\sigma \kappa'$ for the S2T4P3 scheme under the CFL number $\sigma = 0.783$; thick line in (c): track of $Z = -I \sigma \kappa'$ for the S3T5P3 scheme with uniformly partitioned sub-cells and the weight parameter $\gamma = 3$ under the CFL number $\sigma = 0.434$.
Figure 9
Illustration of the secondary reconstruction procedure. The solid lines are primary reconstruction polynomials for each main cell; the dashed lines are secondary reconstruction polynomials extended to $\mathrm{CV}_i$ from two neighboring main cells. (a) Smooth region; (b) near discontinuity.
Figure 10
Comparison of the error for the SCFV schemes and the corresponding order $k$-exact FV schemes. (a) L2 error vs. 1/$h$, with $u_0 (x)=\mathrm{sin}(\pi x)$; (b) L2 error vs. CPU time, with $u_0 (x)=\mathrm{sin}(\pi x)$; (c) L2 error vs. 1/$h$, with $u_0 (x)=\mathrm{sin}^4(\pi x)$; (d) L2 error vs. time, with $u_0 (x)=\mathrm{sin}^4(\pi x)$.
Figure 11
Solution for linear advection problems at $t=16$ using CFL number $\sigma = 0.4$. S2T4P2, with $\gamma=3.00$ and $\lambda_0=100$.
Figure 12
Solution for linear advection problems at $t=16$ using CFL number $\sigma = 0.4$. S2T4P3 with $\lambda_0=100$.
Figure 13
Numerical results for Sod problem at time 0.25 with CFL number $\sigma = 0.4$. $h=1/100$. (a) Third order S2T4P2 scheme, with $\gamma=3.00$ and $\lambda_0=100$; (b) fourth order S2T4P3 scheme, with $\lambda_0=100$.
Figure 14
Numerical results for Lax problem at time 0.1 with CFL number $\sigma = 0.4$. $h=1/200$. (a) Third order S2T4P2 scheme, with $\gamma=3.00$ and $\lambda_0=100$; (b) fourth order S2T4P3 scheme, with $\lambda_0=100$.
Figure 15
The density distribution for Shu-Osher problem at $t=1.8$ with CFL number $\sigma = 0.4$. Square: S2T4P2 scheme with $\gamma=3.00$ and $\lambda_0=100$, $h=1/20$; triangle: fourth order S2T4P3 scheme, with $\lambda_0=100$, $h=1/20$; the solid line: fifth order WENO scheme with $h=1/500$. (a) Density solutions; (b) enlargement of density solutions.
Figure 16
The density distribution for two interaction blast waves at $t=0.038$ with CFL number $\sigma = 0.4$. Square: S2T4P2 scheme with $\gamma=3.00$ and $\lambda_0=100$, $h=1/200$; triangle: fourth order S2T4P3 scheme, with $\lambda_0=1000$, $h=1/200$; the solid line: fifth order WENO scheme with $h$=1/2000.
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