Finite Volume-finite Difference Research Articles

One-dimensional numerical model of landslide over high-steep slopes: Theory and verification to the analytical solution

At the steep slope areas, landslide events may suddenly occur due to prolonged heavy rain and seriously affect humans’ lives and their properties. Numerical models simulating landslides help us to predict the real landslide phenomenon effectively because they are not only time-saving compared to physical models but also acceptable accuracy. In this paper, the one-dimensional nonlinear shallow water equations using the global coordinate system are established as the proposed model governing equations. The hybrid finite difference-finite volume method, the implicit method and the automatical time step controlling technique are employed to stably simulate the landslide events over the high-steep slopes. To validate the proposed numerical model to the analytical solution, authors continuously derive the non-dimensional governing equations in one dimension as well as using the global coordinate system. The numerical results are in good agreement with the analytical solution. The proposed numerical model can be used to simulate the landslides and the landslide-induced tsunami events, especially for simulating the high-steep slope conditions.

Modeling positive surge propagation in open channels using the Serre-Green-Naghdi equations

Numerical modeling of upstream propagation of a positive surge wave in an open-channel flow caused by a sudden rise of downstream water level, either by gate operation or by tidal influence, remains an important challenge from many aspects. A perusal of literature suggests that the Serre-Green-Naghdi (SGN) models have not been methodically explored for a positive surge flow modeling. In this study, the SGN equations for variable bed profiles are solved for simulating the positive surge waves. The governing equations are reformulated in the form of a conservation law, which is solved using a hybrid finite volume-finite difference method. The eddy-viscosity type wave breaking model is coupled with the governing equations for simulating a breaking surge. The performance of second- and fourth-order time stepping schemes and second- and fourth-order accurate schemes for determining the numerical flux is assessed. The results indicate that using an overall fourth-order accurate scheme is preferred for an accurate depiction of the positive surge waves. A comparison is also made with laboratory observations from a wide range of experiments conducted under different discharge, Froude number and bed slope conditions, which demonstrates that the proposed model can accurately reproduce the key free surface characteristics corresponding to positive surge waves. The importance of computing an accurate initial gradually varied flow profile before applying the transient downstream condition, such as a gate closure, is highlighted. Further, the significance of considering full non-linearity is demonstrated through numerical tests as the weakly non-linear Boussinesq-type equations are found to perform poorly in predicting the secondary undulations for an undular surge. The non-hydrostatic pressure field occurring during the propagation of an undular surge has rarely been reported from laboratory experiments. In this study, the bed pressure beneath a propagating undular surge is measured and compared with that computed according to the SGN equations. Finally, the results obtained from the proposed SGN model are compared with those obtained from Large Eddy Simulation results available in the literature.

A high-order implicit least square-based finite difference-finite volume method for incompressible flows on unstructured grids

In this study, a high-order implicit least squares-based finite difference-finite volume (ILSFD-FV) method with a lattice Boltzmann flux solver is presented for the simulation of two-dimensional incompressible flows on unstructured grids. In this method, a high-order polynomial based on Taylor series expansion is applied within each control cell, where the unknown spatial derivatives at each cell center are approximated by the least squares-based finite difference scheme. The volume integral of the high-order polynomial over the control cell results in a pre-multiplied coefficient matrix in the time-dependent term. This makes the high-order method be implicit in nature. With this feature, a high-order implicit Runge–Kutta time integration scheme, namely, the explicit first-stage singly diagonally implicit Runge–Kutta (ESDIRK) scheme, is applied to obtain the time-accurate solutions for flow problems. The non-linear system of equations arising from each ESDIRK stage except for the first explicit stage is solved by a dual time stepping approach. A matrix-free lower-upper symmetric Gauss–Seidel solver is then used to efficiently march the solution in the pseudo time. The present high-order ILSFD-FV method is verified and validated by both steady and unsteady 2D incompressible flow problems. Numerical results indicate that the developed implicit method outperforms its explicit counterpart in terms of the convergence property and computational efficiency. The speedup ratio of the computational effort is about 3–22.

An efficient high-order least square-based finite difference-finite volume method for solution of compressible Navier-Stokes equations on unstructured grids

This paper presents an efficient high-order finite volume method for solution of compressible Navier-Stokes equations on unstructured grids. In this method, a high-order polynomial which is based on Taylor series expansion is applied to approximate the solution function within each control cell. The derivatives in the Taylor series expansion are approximated by the functional values at the cell centers of the considered cell and its neighboring cells using the mesh-free least square-based finite difference (LSFD) scheme. Naturally, this least square-based finite difference-finite volume (LSFD-FV) method inherits appealing characteristics of the LSFD scheme, i.e., the simple algorithm and easy implementation. In addition, since the LSFD scheme is mesh-free in nature, the developed high-order LSFD-FV method is endowed with the flexibility to handle the multi-dimensional problems with complex geometries on arbitrary grids. Different from other high-order methods, this LSFD-FV method applies a novel and effective strategy, i.e., the discrete gas-kinetic flux solver (DGKFS), to compute numerical fluxes at the cell interface. In this way, the inviscid and viscous fluxes are simultaneously and effectively evaluated by local reconstruction of solutions for the Boltzmann equation. The efficient time marching strategy coupled with the high-order LSFD-FV method is adopted to solve the resultant ordinary differential equations with the matching accuracy. A series of numerical examples are presented to validate the accuracy, robustness and flexibility of the developed method on unstructured grids. Numerical results demonstrate the superior performance of the developed high-order method on simulating compressible viscous flows, in comparison with the low-order counterpart and the k-exact method of the same order of accuracy.

Development of an Advection-diffusion Model Using Depth-integrated Equations Based on GPU Acceleration

A scalar transport model is developed by adding a depth-averaged advection-diffusion equation to Celeris Advent, which is a Boussinesq-type numerical model that utilizes GPU acceleration. The hybrid finite volume-finite difference method is used to guarantee numerical stability along with the high accuracy of the Boussinesq equation. The advective and diffusive terms are numerically discretized using the finite volume and finite difference methods, respectively. Results of a one-dimensional scalar advection benchmark test showed that the scalar advection by the proposed model was very close to the analytical solution without any remarkable numerical diffusion. In addition, two benchmark tests using experimental data from different hydraulic experiments were numerically reproduced, and the computed results and observed data for scalar transport were found to be in good agreement. The developed model is expected to contribute to real-time disaster prediction for contaminant spills and can assist in preparing countermeasures for these types of disasters.

A Multi-Dimensional Shock-Capturing Limiter for High-Order Least Square-Based Finite Difference-Finite Volume Method on Unstructured Grids

A Multi-Dimensional Shock-Capturing Limiter for High-Order Least Square-Based Finite Difference-Finite Volume Method on Unstructured Grids

Three-dimensional high-order least square-based finite difference-finite volume method on unstructured grids

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

Predicting deep water wave breaking with a non-hydrostatic shock-capturing model

A non-hydrostatic shock-capturing model is developed to address deep water wave evolution including wave breaking. The present model is based on the Euler equations, with a hybrid finite volume/finite difference (FV/FD) method and the high-resolution Godunov-type numerical scheme. Using these numerical schemes, wave breaking can be captured as a shock or a hydraulic jump without adopting additional wave breaking parameters or terms, such as breaking-onset criteria. The parallel method of the Message Passing Interface (MPI) is used to accelerate the simulation. This model was applied to simulate deep-water wave propagation including breaking as a consequence of dispersive focusing or due to the modulational instability. By comparison with existing experiments, the ability of the present model to predict breaking onset and breaking energy dissipation is evaluated. Additionally, a dynamic criterion is applied to provide additional information on wave breaking onset, and the wavelet transform is applied to predict the energy as a function of frequency and time.

Nearshore coastal flow processes using weighted-averaged equations

This work uses the weighted residual method to simulate non-hydrostatic flows in ocean and coastal areas in a vertically-averaged framework. A Vertically-Averaged and Moment equations set is developed. The system is solved through a hybrid finite volume-finite difference numerical scheme to tackle the hyperbolic and elliptic parts of the equations. Data of seven challenging tests are used to evaluate the model ability to reproduce non-linearity, dispersion, wave breaking, bore propagation, sheet flow and wave reflection. The tests comprise propagation of sinusoidal waves over a submerged bar; convergence of the numerical scheme; solitary wave transformation over a dry reef flat, a wet reef flat and an exposed reef crest; collision of two solitary waves; and solitary wave reflection on a vertical wall under non-breaking and breaking conditions. The results highlight the accuracy of the model without prescribing any treatment for wave breaking. The conjunction of long and short waves with wave shoaling, run-up and receding flows is accurately reproduced. The model is considered an alternative tool to Boussinesq-type models to solve non-linear flows in the nearshore region, with the advantage of having an automatic mimic of wave breaking due to the field variables used to produce the weighted residual depth-averaged equations.

Wave Fields and Nearshore Currents in the Coastal Region Opposite San Mauro Cilento (Italy)

In this paper in order to simulate nearshore currents in computational domains representing the complex morphology of real coastal regions we use a model based on a contravariant integral form of the fully nonlinear Boussinesq equations (FNBE). The contravariant integral form, in which Christoffel symbols are absent, of the continuity equation does not contain the dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations (NSWE). The capacity of the proposed model to correctly simulate the wave train propagation on a highly distorted grid is verified against test case present in the literature. The simulation of wave fields and nearshore currents in the coastal region, opposite San Mauro Cilento (Italy) in presence of a system of T-head groins, is numerically reproduced by using the proposed model.

High-order least-square-based finite-difference-finite-volume method for simulation of incompressible thermal flows on arbitrary grids.

In this work, a high-order (HO) least-square-based finite difference-finite volume (LSFD-FV) method together with thermal lattice Boltzmann flux solver (TLBFS) is presented for simulation of two-dimensional (2D) incompressible thermal flows on arbitrary grids. In the present method, a HO polynomial based on Taylor series expansion is applied within each control cell, where the unknown spatial derivatives at each cell center are approximated by least-square-based finite difference (LSFD) scheme. Then the recently developed TLBFS is applied to evaluate the convective and diffusive fluxes simultaneously at the cell interface by local reconstruction of thermal lattice Boltzmann solutions of the density and internal energy distribution functions. The present HO LSFD-FV method is verified and validated by 2D incompressible heat transfer problems. Numerical results indicate that the present method can be effectively and flexibly applied to solve thermal flow problems with curved boundaries on arbitrary grids. Compared with the conventional low-order finite volume method, higher efficiency and lower memory cost make the present HO method more promising for practical thermal flow problems.

A high order least square-based finite difference-finite volume method with lattice Boltzmann flux solver for simulation of incompressible flows on unstructured grids

This paper presents a high order least square-based finite difference-finite volume (LSFD-FV) method together with lattice Boltzmann flux solver for accurate simulation of incompressible flows on unstructured grids. Within each control cell, a high order polynomial, which is based on Taylor series expansion, is applied to approximate the solution function. The derivatives in the Taylor series expansion are approximated by the functional values at the centers of neighboring cells using the mesh-free least square-based finite difference (LSFD) method developed by Ding et al. (2004) [36]. In the high order finite volume method, the recently developed lattice Boltzmann flux solver (LBFS) is applied to evaluate the inviscid and viscous fluxes physically and simultaneously at the cell interface by local reconstruction of lattice Boltzmann solution. Compared with traditional k-exact high-order finite volume method, the present method is more accurate and efficient. Various benchmark examples are tested to validate the high-order accuracy, high computational efficiency and flexibility of the proposed method on unstructured grids.

Vertically-averaged and moment equations for flow and sediment transport

Simulation of river flow processes including sediment transport is usually conducted using the shallow water flow equations, which are based on a hydrostatic pressure distribution. To increase the accuracy of predictions in a variety of scenarios involving horizontal length scales of the order of vertical length scales, an improved representation of the vertical flow structure is necessary. The mathematical approximation to field variables like the velocity and fluid pressure must be enhanced during the depth-integrating process. Therefore, this paper presents a 1D non-hydrostatic flow and sediment transport model developed by using the method of the weighted residuals into the RANS equations. Using continuity, momentum, and moment equations, the fluid pressure distribution is modelled using a quadratic predictor with perturbation parameters to deviate the vertical momentum balance from the hydrostatic law. The flow equations are a generalized non-hydrostatic flow solver, where the fluid density variation due to suspension of sediments and the bed deformation due to erosion-sedimentation processes are accounted for. A hybrid semi-implicit finite volume-finite difference numerical scheme is developed to solve the system of conservation laws. Two different approaches are used to model the sediment transport processes: (i) Unified computation of the total-load transport and (ii) separate computation of suspended and bed loads. The accuracy of the non-hydrostatic model is demonstrated by comparison with experimental data, highlighting better results accounting for separate determinations of the suspended and bed loads in highly erosive flows.

Celeris: A GPU-accelerated open source software with a Boussinesq-type wave solver for real-time interactive simulation and visualization

In this paper, we introduce an interactive coastal wave simulation and visualization software, called Celeris. Celeris is an open source software which needs minimum preparation to run on a Windows machine. The software solves the extended Boussinesq equations using a hybrid finite volume–finite difference method and supports moving shoreline boundaries. The simulation and visualization are performed on the GPU using Direct3D libraries, which enables the software to run faster than real-time. Celeris provides a first-of-its-kind interactive modeling platform for coastal wave applications and it supports simultaneous visualization with both photorealistic and colormapped rendering capabilities. We validate our software through comparison with three standard benchmarks for non-breaking and breaking waves. Program summaryProgram title: CelerisProgram Files doi:http://dx.doi.org/10.17632/5djwvf5x5k.1Licensing provisions : GNU General Public License 3 (GPL)Programming language : C++, HLSLNature of problem : Boussinesq-type models provide the research-level accuracy needed for modeling wave propagation in coastal zones. However the current models, both commercial and open source, do not provide means for real-time computation, nor provide model interactivity and concurrent visualization. In order to achieve a real-time simulation speed in current parallelized models, dozens to hundreds of CPU cores are needed. Celeris is an interactive software which provides faster than real-time simulation and visualization speed on an average user laptop. The novelty of this software is its interactive environment, which allows the user to modify the model and field parameters as the model is running, and to see the effect of these changes immediately.Solution method : A hybrid finite volume-finite difference scheme is used to solve the extended Boussinesq equations. The solver is parallelized using shader programming with Direct3D libraries. Visualization is also performed with the same libraries.

Ritter’s dry-bed dam-break flows: positive and negative wave dynamics

Dam-break flood waves are associated with major environmental disasters provoked by the sudden release of water stored in reservoirs. Ritter found in 1892 an analytical solution to the wave structure of an ideal fluid released during an instantaneous dam failure, propagating over initially dry horizontal terrain. This solution, though ideal, hence frictionless, is widely used to test numerical solutions of the Shallow Water Equations (SWE), and as educational tool in courses of fluid mechanics, given that it is a peculiar case of the Riemann problem. However, the real wave structure observed experimentally differs in a major portion of the wave profile, including the positive and negative fronts. Given the importance of an accurate prediction of the dam break wave, the positive and negative wave portions originating from the breaking of a dam with initially dry land on the tailwater reach are revisited in this work. First, the propagation features of the dry-front are investigated using an analytical boundary-layer type model (Whitham/Dressler/Chanson model) constructed matching an (outer) inviscid dynamic wave to an (inner) viscous diffusive wave. The analytical solution is evaluated using an accurate numerical solution of the SWE produced using the MUSCL-Hancock finite-volume method, which is tested independently obtaining the solution based on the discontinuous Galerkin finite-element method. The propagation features of the negative wave are poorly reproduced by the SWE during the initial stages of dam break flows, and, thus, are then investigated using the Serre–Green–Naghdi equations for weakly-dispersive fully non-linear water waves, which are solved using a finite volume-finite difference scheme.

Numerical simulation of wave transformation, breaking and runup by a contravariant fully non-linear Boussinesq equations model

In this paper we propose a new model based on a contravariant integral form of the fully non-linear Boussinesq equations (FNBE) in order to simulate wave transformation phenomena, wave breaking, runup and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. A high-order upwind weighted essentially non-oscillatory (WENO) finite volume scheme that involves an exact Riemann solver is implemented. The wave breaking is represented by discontinuities of the weak solution of the integral form of the non-linear shallow water equations (NSWE). On the basis of the shock-capturing high order WENO scheme a new procedure, for the computation of the structure of the solution of a Riemann problem associated with a wet/dry front, is proposed in order to simulate the run up hydrodynamics in swash zone. The capacity of the proposed model to correctly represent wave propagation, wave breaking, run up and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. The proposed model is applied to a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite San Mauro Cilento (Italy).

A high-order hybrid finite difference–finite volume approach with application to inviscid compressible flow problems: A preliminary study

A class of hybrid finite difference–finite volume (FD–FV) operators is recently developed as building blocks to solve one dimensional hyperbolic conservation laws when the solutions are smooth. This method differs from conventional finite difference (FD) or finite volume (FV) schemes in that both nodal values and cell-averaged values are considered as dependent variables and they are evolved in time. Under this framework, the 1D FD–FV methods: (1) are numerically conservative for cell averages; (2) have straightforward extension to high-order accuracy; and (3) have superior spatial accuracy property compared to most conventional FD or FV methods.This work extends the FD–FV approach in two aspects. The first extension is a WENO-type stabilization to enhance the nonlinear stability of sample high-order 1D FD–FV operators. In particular, numerical results show that when the solutions are smooth, the optimal order of accuracy (fifth-order) is achieved by the stabilized fourth-order FD–FV method; and it is also capable to handle problems with strongly discontinuous solutions.The second part of the paper extends a second-order FD–FV method to two-dimensional smooth problems. Both Cartesian grids and unstructured (triangular) grids are considered. In multiple dimensions, there are different choices of the collocation points of the nodal values, and they lead to different FD–FV schemes. This work develops a node-centered FD–FV scheme and an edge-centered FD–FV scheme on each type of grids, and their numerical performance are assessed and compared by solving benchmark flow problems with smooth solutions. In particular, the numerical examples confirm that the superior spatial accuracy property of the 1D FD–FV operators carries to two space dimensions on Cartesian grids. The present work focuses on two space dimensions, but the methodology extends naturally to three-dimensional Cartesian grids and tetrahedral grids.

Numerical Solution of Inviscid Transonic Flow Using Hybrid Finite Volume-Finite Difference Solution Technique on Unstructured Grid

An explicit cell centered finite volume CFD solver called Pyramid2D has been developed for solving Euler’s equations in two dimensions. We modify the flow equations at the boundary faces, according to the physical boundary conditions, and solve them in finite difference discrete form suitable for unstructured grids. The flow properties at the cell faces are constructed using new reconstruction method which allows small oscillations at the shock wave region and in the same time contains enough diffusion, so it doesn’t require the use of limiters for transonic flows. The developed solver is to be used as an external aerodynamic solver; all tests showed that the proposed scheme is robust and its numerical results compares well with the published data.

C3: A finite volume-finite difference hybrid model for tsunami propagation and runup

A numerical model that couples Finite Difference and Finite Volume schemes has been developed for tsunami propagation and runup study. An explicit leap-frog scheme and a first order upwind scheme has been considered in the Finite Difference module, while in the Finite Volume scheme a Godunov Type method based on the f-waves approach has been used. The Riemann solver included in the model corresponds to an approximate augmented solver for the Shallow Water Equations (SWE) in the presence of variable bottom surface. With this hybrid model some of the problems inherent to the Godunov type schemes are avoided in the offshore region, while in the coastal area the use of a conservative method ensures the correct computation of the runup and wave breaking. The model has been tested and validated using different problems with a known analytical solution and also with laboratory experiments, considering both non breaking and breaking waves. The results are very satisfactory, showing that the hybrid approach is a useful technique for practical usages.

Finite volume scheme for the solution of 2D extended Boussinesq equations in the surf zone

In this paper, a hybrid finite volume-finite difference scheme is applied to study surf zone dynamics. The numerical model solves the 2DH extended Boussinesq equations proposed by Madsen and Sørensen (1992) where nonlinear and dispersive effects are both relevant whereas it solves NSWE equations where nonlinearity prevails. The shock-capturing features of the finite volume method allow an intrinsic representation of wave breaking and runup; therefore no empirical (calibration) parameters are necessary. Comparison with laboratory measurements demonstrates that the proposed model can accurately predict wave height decay and mean water level setup, for both regular and solitary wave breaking on a sloping beach. The model is also applied to reproduce two-dimensional wave transformation and breaking over a submerged circular shoal, showing good agreement with experimental data.