Finite Volume Difference Scheme Research Articles

Scalable implicit solvers with dynamic mesh adaptation for a relativistic drift-kinetic Fokker–Planck–Boltzmann model

In this work we consider a relativistic drift-kinetic model for runaway electrons along with a Fokker–Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source. We develop a new scalable fully implicit solver utilizing finite volume and conservative finite difference schemes and dynamic mesh adaptivity. A new data management framework in the PETSc library based on the p4est library is developed to enable simulations with dynamic adaptive mesh refinement (AMR), distributed memory parallelization, and dynamic load balancing of computational work. This framework and the runaway electron solver building on the framework are able to dynamically capture both bulk Maxwellian at the low-energy region and a runaway tail at the high-energy region. To effectively capture features via the AMR algorithm, a new AMR indicator prediction strategy is proposed that is performed alongside the implicit time evolution of the solution. This strategy is complemented by the introduction of computationally cheap feature-based AMR indicators that are analyzed theoretically. Numerical results quantify the advantages of the prediction strategy in better capturing features compared with nonpredictive strategies; and we demonstrate trade-offs regarding computational costs. The robustness with respect to model parameters, algorithmic scalability, and parallel scalability are demonstrated through several benchmark problems including manufactured solutions and solutions of different physics models. We focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.

Advancements in nearshore wave modeling: A unified one-layer nonhydrostatic approach

This work presents a novel one-layer nonhydrostatic formulation and model for nearshore waves. The proposed governing equations define velocities and pressures at arbitrary distances from the still water and only contain spatial derivatives of maximum second order. The formulation can be unified into the existing nonhydrostatic models by defining the variables at the middle depth and neglecting certain additional terms. A Stokes-type Fourier analysis was performed to analyze the formulations' properties and determine the location of variables. The proposed formulation exhibited a clear superiority in describing both the linear and nonlinear properties of the coastal waves. The equations were numerically solved using a hybrid-finite, volume-finite difference scheme. The resulting model accurately described the wave-breaking and runup processes that occurred due to the adoption of a shock-capturing scheme and seabed elevation reconstruction. The suggested novel numerical model was validated against two theoretical benchmark tests and three wave transformation experiments.

New finite difference unequal-sized Hermite WENO scheme for Navier-Stokes equations

In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spatial reconstruction procedure uses a convex combination of one quintic polynomial and two linear polynomials defined on three unequal-sized central or biased spatial stencils. Compared with the classical finite difference, finite volume, or mixed finite difference and finite volume HWENO schemes [32] , [33] , the new scheme uses same largest stencil and only three stencils in spatial reconstruction procedures, and the first-order derivative values in auxiliary equations can be obtained directly rather than using complex HWENO-type reconstruction methodologies [32] , [33] . Moreover, the associated viscosity terms could be directly computed from the function values and first-order derivative values by using the characteristic of the WENO-type spatial reconstructions. Meanwhile, the linear weights can be any positive numbers on condition that their summation equals one, which avoids the problem of negative linear weights in classical HWENO schemes [24] , [25] . So the new finite difference US-HWENO scheme is more suitable for solving Navier-Stokes equations, and can be easily implemented to multi-dimensions on structured meshes. Some benchmark viscous examples are illustrated to verify the good performance of this new finite difference US-HWENO scheme.

Positivity-preserving finite volume difference schemes for atmospheric dispersion models with degenerate vertical diffusion

Positivity-preserving finite volume difference schemes for atmospheric dispersion models with degenerate vertical diffusion

A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential

The hydrostatic equilibrium state is the consequence of the exact hydrostatic balance between hydrostatic pressure and external force. Standard finite volume or finite difference schemes cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibrium state and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations with a global reconstruction. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested numerically to demonstrate the accuracy and the stability of the new scheme, and the results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. The viscous effect is also illustrated through the propagation of small perturbation and the Rayleigh-Taylor instability. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibrium state from a highly non-balanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters

In this paper, a new type of multi-resolution weighted essentially non-oscillatory (WENO) limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods is designed. This type of multi-resolution WENO limiters is an extension of the multi-resolution WENO finite volume and finite difference schemes developed in [43]. Such new limiters use information of the DG solution essentially only within the troubled cell itself, to build a sequence of hierarchical L2 projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, fourth-order, and fifth-order RKDG methods with these multi-resolution WENO limiters have been developed as examples, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original DG methods. Benchmark examples are given to demonstrate the good performance of these RKDG methods with the associated multi-resolution WENO limiters.

Celeris Base: An interactive and immersive Boussinesq-type nearshore wave simulation software

We introduce our interactive and immersive coastal wave simulation software, Celeris Base, which is the successor to Celeris Advent. Celeris Base is an open source software developed in the Unity3D game engine and in C# language. It supports an interactive environment and allows users to view the simulations in a virtual reality headset. Celeris Base solves the same equations as Celeris Advent, the extended Boussinesq equations, using our hybrid finite volume–finite difference method. These equations are solved on the GPU using compute shaders, written in HLSL. Celeris Base has several new features such as 360°video capturing, geographic map overlays, built-in real-time gauge plotters, etc. It also improves the implementation of the sponge layer boundary condition by introducing new damping equations. Celeris Base is designed and implemented using the best software engineering practices in the hope that it will be a base for further developments of the Celeris software series by researchers around the globe. We validate Celeris Base against experimental results in this paper. Program summaryProgram Title: Celeris BaseProgram Files doi:http://dx.doi.org/10.17632/jdx7tddcxz.1Licensing provisions: MIT LicenseProgramming language: C#, HLSLNature of problem: Celeris Advent enabled researchers and engineers for the first time to simulate nearshore waves with a Boussinesq-type model, faster than real-time and in an interactive environment. However, its development platform and implementation complexity hindered researchers from developing it further and made adding new features to the software a daunting task. The software used graphics shaders to solve scientific equations which could be confusing for many. The visualization environment was wired from scratch which made it very difficult to add features such as virtual reality.Solution method: A new software is developed completely from scratch following Celeris Advent, called Celeris Base. This software uses the same hybrid finite volume–finite difference scheme to solve the extended Boussinesq equations, but using a variant of shaders called compute shaders, removing possible barriers for other researchers to understand the code and develop it further. The software is developed in Unity3D, a popular game engine with a large and helpful community as well as thousands of ready to use plugins. Celeris Base is equipped with virtual reality and is the first nearshore simulation software to provide this feature.

Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Central WENO reconstruction procedures have shown very good performances in finite volume and finite difference schemes for hyperbolic conservation and balance laws in one and more space dimensions, on different types of meshes. Their most recent formulations include WENOZ-type nonlinear weights, but in this context a thorough analysis of the global smoothness indicator $\tau$ is still lacking. In this work we first prove results on the asymptotic expansion of one- and multi-dimensional Jiang-Shu smoothness indicators that are useful for the rigorous design of a CWENOZ schemes, also beyond those considered in this paper. Next, we introduce the optimal definition of $\tau$ for the one-dimensional CWENOZ schemes and for one example of two-dimensional CWENOZ reconstruction. Numerical experiments of one and two dimensional test problems show the correctness of the analysis and the good performance of the new schemes.

Numerical Simulation of Wave Propagation, Breaking, and Setup on Steep Fringing Reefs

The prediction of wave transformation and associated hydrodynamics is essential in the design and construction of reef top structures on fringing reefs. To simulate the transformation process with better accuracy and time efficiency, a shock-capturing numerical model based on the extended Boussinesq equations suitable for rapidly varying topography with respect to wave transformation, breaking and runup, is established. A hybrid finite volume–finite difference scheme is used to discretize conservation form of the extended Boussinesq equations. The finite-volume method with a HLL Riemann solver is applied to the flux terms, while finite-difference discretization is applied to the remaining terms. The fourth-order MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) scheme is employed to create interface variables, with in which the van-Leer limiter is adopted to improve computational accuracy on complex topography. Taking advantage of van-Leer limiter, a nested model is used to take account of both computational run time and accuracy. A modified eddy viscosity model is applied to better accommodate wave breaking on steep reef slopes. The established model is validated with laboratory measurements of regular and irregular wave transformation and breaking on steep fringing reefs. Results show the model can provide satisfactory predictions of wave height, mean water level and the generation of higher harmonics.

Non-hydrostatic pressure shallow flows: GPU implementation using finite volume and finite difference scheme

We consider the depth-integrated non-hydrostatic system derived by Yamazaki et al. An efficient formally second-order well-balanced hybrid finite volume finite difference numerical scheme is proposed. The scheme consists of a two-step algorithm based on a projection-correction type scheme initially introduced by Chorin–Temam [15]. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite volume method. Second, the dispersive terms are solved by means of compact finite differences. A new methodology is also presented to handle wave breaking over complex bathymetries. This adapts well to GPU-architectures and guidelines about its GPU implementation are introduced. The method has been applied to idealized and challenging experimental test cases, which shows the efficiency and accuracy of the method.

Heat transfer in an enclosure filled with a nanofluid and containing a heat-generating conductive body

This paper performs a numerical analysis of the conjugate heat transfer within a square enclosure full of a copper-water nanofluid. The enclosure also contains a heat-generating solid square block (a heat source) at the center. While the horizontal walls of the enclosure are considered adiabatic, its vertical walls are maintained at a constant low temperature. A control volume finite difference scheme, along with the SIMPLE algorithm, is adopted to conduct the numerical analyses and solve the discrete equations. The results of the numerical analyses are then used to clarify the effects of relevant factors, e.g. Rayleigh number, solid volume fraction, thermal conductivity of the heat source and length ratio on the flow field and heat transfer. The results are reported in terms of streamlines, isotherms, velocity and temperature profiles across the enclosure, and local and average Nusselt numbers. According to my findings, augmentation of Rayleigh number and the solid volume fraction enhance the thermal performance of the enclosure. Moreover, the increase in the length ratio affects the rate of heat transfer. Finally, higher thermal conductivity of the heat source is associated with increments in the temperature of the nanofluid in the enclosure, the temperature of the heat source, and the average Nu.

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax–Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes.

A new numerical model for simulations of wave transformation, breaking and long‐shore currents in complex coastal regions

SummaryIn this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations.The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.

Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation

The Schrödinger equation is a model for many physical processes in quantum physics. It is a singularly perturbed differential equation where the presence of the small reduced Planck's constant makes the classical numerical methods very costly and inefficient. We design two new schemes. The first scheme is the nonstandard finite volume method, whereby the perturbation term is approximated by nonstandard technique, the potential is approximated by its mean value on the cell and the complex dependent boundary conditions are handled by exact schemes. In the second scheme, the deficiency of classical schemes is corrected by the inner expansion in the boundary layer region. Numerical simulations supporting the performance of the schemes are presented.

A discontinuous Galerkin scheme for Chimera overset viscous meshes on curved geometries

The Chimera overset method is a powerful technique for modeling fluid flow associated with complex engineering problems. The use of structured meshes has enabled engineers to develop a number of high-order schemes, such as the WENO and compact differencing schemes. This paper demonstrates a Discontinuous Galerkin (DG) scheme with a Chimera overset method applied to viscous meshes on curved geometries. The small stencil of the DG scheme makes it particularly suitable for Chimera meshes. The small stencil simplifies the hole cutting and partitioning of grids that contain holes. In addition, because the DG scheme represents the solution as cell local polynomials, it does not require an interpolation scheme with a large stencil to establish the inter-grid communication in overlapping regions. Furthermore, the DG scheme is capable of using curved cells to represent geometric features. The curved cells resolve issues associated with linear non-co-located Chimera viscous meshes used for finite volume and finite difference schemes. The DG-Chimera method is demonstrated on a set of viscous Chimera meshes, which would produce erroneous results for a finite volume or finite difference scheme without corrections to the interpolation.

A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations

A total variation diminishing-weighted average flux (TVD-WAF)-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer (HLL) Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth-order monotone upstream-centered scheme for conservation laws (MUSCL). The time marching scheme based on the third-order TVD Runge-Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.

Fitted finite volume positive difference scheme for a stationary model of air pollution

A new approach is proposed for the numerical solution of boundary value one-dimensional problem of advection-diffusion equation which arises, among others, in air pollution modeling. Since the problem is posed in an unbounded interval we use a log-transformation to confine the computational region. We discuss the well-posedness of the new problem and the properties of its solution. We derive a positive finite volume difference scheme. Some results from computational experiments are presented.

A finite volume difference scheme for a model of settling particle dispersion from an elevated source in an open-channel flow

A finite volume fitted difference scheme is constructed to solve the unsteady convective-diffusion equation transformed on a finite domain, modeling longitudinal dispersion of suspended particles with settling velocity in a turbulent shear flow over a rough-bed surface. First we discuss the well-posedness of the differential problem and the non-negativity of its solution. Then, to overcome the degeneracy at the part of the boundary, starting from the divergent form of the equation we perform a local fitted space discretization. This approximation is determined by a set of two-point boundary value problems. Non-negativity of the numerical concentration of suspended fine particles is proved. Some results from computational experiments are presented to illustrate the properties of the constructed scheme.

An integral contravariant formulation of the fully non-linear Boussinesq equations

In this paper we propose an integral form of the fully non-linear Boussinesq equations in contravariant formulation, in which Christoffel symbols are avoided, in order to simulate wave transformation phenomena, wave breaking and nearshore currents in computational domains representing the complex morphology of real coastal regions. Following the approach proposed by Chen (2006), the motion equations retain the term related to the approximation to the second order of the vertical vorticity. A new Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the fully non-linear Boussinesq equations on generalised curvilinear coordinate systems is proposed. The equations are rearranged in order to solve them by a high resolution hybrid finite volume–finite difference scheme. The conservative part of the above-mentioned equations, consisting of the convective terms and the terms related to the free surface elevation, is discretised by a high-order shock-capturing finite volume scheme in which an exact Riemann solver is involved; dispersive terms and the term related to the approximation to the second order of the vertical vorticity are discretised by a cell-centred finite difference scheme. The shock-capturing method makes it possible to intrinsically model the wave breaking, therefore no additional terms are needed to take into account the breaking related energy dissipation in the surf zone. The model is verified against several benchmark tests, and the results are compared with experimental, theoretical and alternative numerical solutions.