Compact Finite Volume Scheme Research Articles

A third-order compact finite volume scheme on unstructured grid for fluid flows

The large reconstruction stencil is one of the biggest obstacles in design and implementation of high-order finite volume schemes on unstructured grid. To tackle this problem, a third-order compact reconstruction scheme on unstructured grid is developed on the basis of compact least-squares (CLS) finite volume method. In building the reconstruction polynomial, the general boundary condition is adopted to ensure the unified third-order accuracy in both the internal and boundary cells. The reconstruction polynomial is solved in a semi-implicit sense to ensure high computational efficiency without solving the global linear algebraic equations. The theoretical accuracy order and stability limit are firstly verified by the Fourier analysis. The numerical accuracy and efficiency are then verified and validated with several incompressible and compressible flow cases. This work provides a reference to design and analysis of high-order compact finite volume schemes on unstructured grid with unified accuracy.

On wave-current interaction with a horizontal cylinder near free surface by a fourth-order-accurate finite volume compact solver

We present a numerical solver for dealing with the problem of wave-current interaction with a horizontal cylinder near free surface. The solver solves the filtered Navier-Stokes equations which are discretized using a fourth-order-accurate compact finite volume scheme on a collocated grid. The turbulence effect is modelled by a subgrid-scale model using a large eddy simulation, and the free surface and the cylinder are modelled by a volume of fluid method and a ghost cell immersed boundary method. The pressure-velocity coupling is ensured via a momentum interpolation method. We obtain the hydrodynamic force and its spectra characteristics, dynamic pressure distribution, streamline topology and vorticity patterns with different immersed depths under uniform currents, regular waves and wave-current flows. Validation is performed by comparing the calculated hydrodynamic force with the measured data within one wave period. The results show that the position of vortex shedding and dissipation is different for the cases under different immersed depth in regular waves and wave-current flows. The dynamic pressure around the cylinder as the force reaches its peak value changes with immersed depth. The value and phase of the force are sensitive to immersed depth.

A Compact Finite Volume Scheme for the Multi-Term Time Fractional Sub-Diffusion Equation

In this paper, we introduce high-order finite volume methods for the multi-term time fractional sub-diffusion equation. The time fractional derivatives are described in Caputo’s sense. By using some operators, we obtain the compact finite volume scheme have high order accuracy. We use a compact operator to deal with spatial direction; then we can get the compact finite volume scheme. It is proved that the finite volume scheme is unconditionally stable and convergent in L∞-norm. The convergence order is O(τ2-α + h4). Finally, two numerical examples are given to confirm the theoretical results. Some tables listed also can explain the stability and convergence of the scheme.

High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation

Based on an L1 interpolation operator, a new high-order compact finite volume scheme is derived for the 2D multi-term time fractional sub-diffusion equation. It is shown that the difference scheme is unconditionally convergent and stable in L_{infty }-norm. The convergence order is O(tau ^{2-alpha }+h_{1}^{4}+h_{2}^{4}), where τ is the temporal step size and h_{1} is the spatial step size in one direction, h_{2} is the spatial step size in another direction. Two numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

A compact finite volume method and its extrapolation for elliptic equations with third boundary conditions

A fourth-order compact finite volume method is constructed for one and two dimensional elliptic equations with third boundary conditions in this paper. Taking two point boundary value problem of third kind as an example, we derive some useful high accuracy post-processing formulas to recover the numerical derivatives at the nodes or midpoints of the elements. We also improve the accuracy of the compact finite volume scheme from order 4 to 6 based on Richardson extrapolation by rigorously proving the scheme has error asymptotic expansion. Numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme as well as its post-processing formulas and extrapolation.

A Padé compact high-order finite volume scheme for nonlinear Schrödinger equations

In this work, a Padé compact high-order finite volume scheme is presented for the solution of one-dimensional nonlinear Schrödinger equations. The compact high-order finite volume schemes posses inherent conservation of the equations and high order accuracy within small stencils. Fourier error analysis demonstrates that the spectral resolution of the Padé compact finite volume scheme exceeds that of the standard finite volume schemes in terms of the same order of accuracy. Besides, the linear stability of the temporal discretization scheme is also performed by using the Fourier analysis. Numerical results are obtained for the nonlinear Schrödinger equations with various initial and boundary conditions, which manifests high accuracy and validity of the Padé compact finite volume scheme.

Fourth order compact finite volume scheme on nonuniform grids with multi-blocking

We have developed a fourth order compact finite volume method for the solution of low Mach number compressible flow equations on arbitrary nonuniform grids. The formulation presented here uses collocated grid that preserves fourth order accuracy on nonuniform meshes. This was achieved by introduction of a new fourth order method for calculation of cell and face averaged metrics. A special treatment of nonlinear terms is used to guarantee the stability of the fourth order compact method. Moreover an approach for applying this method to multi-block domains is presented for complicated geometries and parallel processing applications. Several test cases including the flow in a lid-driven cavity, laminar flow over a flat plate, decay of the Taylor vortex, and transient flow behind a backward facing step show the accuracy, efficiency, and stability of the method in single and multi-block domains. To the best of our knowledge, this is the first symmetric compact method in the finite volume form that can achieve full fourth order accuracy in solving fluid flow equations on nonuniform collocated grids without any stability problems.

A fourth‐order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq‐type equations. Part II: boundary conditions and validation

AbstractThis paper supplements the validation of the fourth‐order compact finite volume Boussinesq‐type model presented by Cienfuegos et al. (Int. J. Numer. Meth. Fluids 2006, in press). We discuss several issues related to the application of the model for realistic wave propagation problems where boundary conditions and uneven bathymetries must be considered. We implement a moving shoreline boundary condition following the lines given by Lynett et al. (Coastal Eng. 2002; 46:89–107), while an absorbing‐generating seaward boundary and an impermeable vertical wall boundary are approximated using a characteristic decomposition of the Serre equations. Using several benchmark tests, both numerical and experimental, we show that the new finite volume model is able to correctly describe nonlinear wave processes from shallow waters and up to wavelengths which correspond to the theoretical deep water limit. The results compare favourably with those reported using former fully nonlinear and weakly dispersive Boussinesq‐type solvers even when time integration is conducted with Courant numbers greater than 1.0. Furthermore, excellent nonlinear performance is observed when numerical computations are compared with several experimental tests on solitary waves shoaling over planar beaches up to breaking. A preliminary test including the wave‐breaking parameterization described by Cienfuegos (Fifth International Symposium on Ocean Wave Measurement Analysis, Madrid, Spain, 2005) shows that the Boussinesq model can be extended to deal with surf zone waves. Finally, practical aspects related to the application of a high‐order implicit filter as given by Gaitonde et al. (Int. J. Numer. Methods Engng 1999; 45:1849–1869) to damp out unphysical wavelengths, and the numerical robustness of the finite volume scheme are also discussed. Copyright © 2006 John Wiley & Sons, Ltd.

A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: model development and analysis

A high-order finite volume scheme is developed to numerically integrate a fully nonlinear and weakly dispersive set of Boussinesq-type equations (the so-called Serre equations) (J. Fluid Mech. 1987; 176:117–134; Surveys Geophys. 2004; 25(3–4):315–337). The choice of this discretization strategy is motivated by the fact that this particular set of equations is recasted in a convenient quasi-conservative form. Cell face values are reconstructed using implicit compact schemes (J. Comput. Phys. 1999; 156:137–180; J. Comput. Phys. 2004; 198:535–566) and time integration is performed with the help of a four-stage Runge–Kutta method. Numerical properties of the proposed scheme are investigated both, analytically using linear spectral analysis, and numerically for highly nonlinear cases. The numerical analysis indicates that the newly developed scheme has wider stability regions and better spectral resolution than most of the previously published numerical methods used to handle equivalent set of equations. Moreover, it was also noticed that the use of mixed-order strategies to discretize convective and dispersive terms may result in an important overall reduction of the spectral resolution of the scheme. Additionally, there is some numerical evidence, which seems to indicate that the incorporation of a high-order dispersion correction term as given by Madsen et al. (Coastal Eng. 1991; 15:371–388) may introduce instability in the system. Copyright © 2006 John Wiley & Sons, Ltd.

Evaluation of a Fourth-Order Finite-Volume Compact Scheme for LES with Explicit Filtering

ABSTRACT The accuracy of the large-eddy simulation (LES) of turbulent flows can be increased by using high-order numerical schemes in space and time, due to a decrease in numerical errors. This work investigates a high-order compact finite-volume scheme suitable for LES. The explicit fourth-order Runge-Kutta (RK) scheme for time marching and fourth-order compact schemes for spatial derivatives using a cell-averaged approach are implemented. Different subgrid-scale models and the effect of explicit filtering in a fully turbulent channel flow are studied. In this flow, the fourth-order compact finite-volume method in space, and fourth-order RK in time in conjunction with the dynamic Smagorinsky model with explicit filter-grid size ratio of , show the best agreement with the available reference data.

A new flux–vector splitting compact finite volume scheme

A new high resolution finite volume method is proposed here that uses flux–vector splitting as the building block to represent the physical processes. The high accuracy projection in space is obtained here, inter alia, by using some compact schemes of Sengupta et al. [1] and Lele [2] – that have extremely high accuracy in spectral plane. This produces a new class of finite volume (FV) scheme for spatial discretization, that has been analyzed for spectral accuracy, numerical stability and dispersion relation preservation (DRP) property following the full-domain analysis method of [1] using two different time integration strategies. These combination of spatial and temporal methods have been tested for two linear wave problems, reported in [1,3] and the solution of Burgers’ equation is compared with the analytical results presented in Adams and Shariff [4]. The shock capturing method is based on the proposed technique in [5], that does not require the usage of any nonlinear differencing techniques or limiters. To show that the proposed method can also handle irregular grids, the Burgers’ equation is solved using non-uniform grids. Also to demonstrate the utility of the proposed scheme to solve practical problems, the benchmark problem of nonlinear wave propagation in a one-dimensional shock-tube has been solved with the initial data as given in [23] by solving the Euler equation. The analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method, at the same time providing a neutrally stable scheme when used with four stage Runge–Kutta scheme. Also the same scheme displays extended ranges of wave numbers and circular frequencies over which DRP property is valid. The computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems.