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This special issue is dedicated to recent developments concerning modern numerical methods for time dependent partial differential equations that can be used to simulate complex nonlinear flow and transport processes and tries to cover a wide spectrum of different applications and numerical approaches.
Most of the models under consideration here are nonlinear hyperbolic conservation laws, which are a very powerful mathematical tool to describe a large set of problems in science and engineering. Hyperbolic PDE are based on first principles, such as mass, momentum and total energy conservation as well as the second law of thermodynamics. These principles are universally valid and have been successfully used for the simulation of a very wide class of problems, ranging from astrophysics over geophysics to complex flows in engineering and computational biology. Most of the seemingly different applications listed above have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations, possibly containing also higher-order derivative terms, non-conservative products and nonlinear (potentially stiff) source terms. From the mathematical point of view, the major difficulties in these systems arise from the nonlinearities that allow the formation of non-smooth solutions such as shock waves, material interfaces or other discontinuities, but nonlinear conservation laws can simultaneously also contain smooth features like sound waves and vortex flows. Even nowadays the key challenge for the construction of successful numerical methods for nonlinear hyperbolic PDE is therefore to capture at the same time shock waves and discontinuities robustly without producing spurious oscillations and exhibit little numerical dissipation and dispersion errors in smooth regions of the solution. For this reason, the development of robust and accurate numerical methods for nonlinear flow and transport problems remains a great challenge even after several decades of successful research, which started back in the 1940s and 1950s with the pioneering works of von Neumann [1] and Godunov [2].
The present special issue is published on the occasion of the 70th birthday of Prof. Eleuterio F. Toro and in honor of his outstanding contributions to the fields of applied mathematics, scientific computing and computational physics with his well-known numerical methods for the solution of hyperbolic conservation laws and their application to computational fluid dynamics, computational physics and computational biology.

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Recent advances in finite-difference WENO schemes for hyperbolic conservation laws have resulted in WENO schemes with adaptive order of accuracy. For instance, a WENO-AO(5,3) scheme can provide up to fifth order of accuracy when the smoothness of the solution in the fifth order stencil warrants it, and yet, it can adaptively drop down to third order of accuracy when the higher order is not warranted by the solution on the mesh. Having an analogous capability for finite-volume WENO schemes for hyperbolic conservation laws, especially on unstructured meshes, can be very valuable. The present paper documents the design of finite volume WENO-AO(4,3) and WENO-AO(5,3) schemes for unstructured meshes. As with WENO-AO for structured meshes, the key advance lies in realizing that there is a favorable basis set, which is very easily constructed, and in which the computation is dramatically simplified. As with finite-difference WENO, we realize that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order central WENO scheme that is, nevertheless, very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes that work well on unstructured meshes. On both the large and small stencils we have been able to make the stencil evaluation step very efficient owing to the choice of a favorable Taylor series basis. By extending the Parallel Axis Theorem, we show that there is a significant simplification in the finite volume reconstruction. Instead of solving a constrained least squares problem, our method only requires the solution of a smaller least squares problem on each stencil. This also simplifies the matrix assembly and solution for each stencil. The evaluation of smoothness indicators is also simplified. Accuracy tests show that the method meets its design accuracy. Several stringent test problems are presented to demonstrate that the method works very robustly and very well. The test problems are chosen to show that our method can be applied to many different meshes that are used to map geometric complexity or solution complexity.

In this paper, we present a novel second-order accurate Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving nonconforming polygonal grids, in order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type methods. In our new approach the nonconforming element interfaces are not defined by the user, but they are automatically detected by the algorithm if the tangential velocity difference across an element interface is sufficiently large. The grid nodes that are sufficiently far away from a shear wave are moved with a standard node solver, while at the interface we insert a new set of nodes that can slide along the interface in a nonconforming manner. In this way, the elements on both sides of the shear wave can move with a different velocity, without producing highly distorted elements.
The core of the proposed method is the use of a space-time conservation formulation in the construction of the final finite volume scheme, which completely avoids the need of an additional remapping stage, hence the new method is a so-called direct ALE scheme. For this purpose, the governing PDE system is rewritten at the aid of the space-time divergence operator and then a fully discrete one-step discretization is obtained by integrating over a set of closed space-time control volumes. The nonconforming sliding of nodes along an edge requires the insertion or the deletion of nodes and edges, and in particular the space-time faces of an element can be shared between more than two cells.
Due to the space-time conservation formulation, the geometric conservation law (GCL) is automatically satisfied by construction, even on moving nonconforming meshes. Moreover, the mesh quality remains high and, as a direct consequence, also the time step remains almost constant in time, even for highly sheared vortex flows. In this paper we focus mainly on logically straight slip-line interfaces, but we show also first results for general slide lines that are not logically straight. Second order of accuracy in space and time is obtained by using a MUSCL-Hancock strategy, together with a Barth and Jespersen slope limiter.
The accuracy of the new scheme has been further improved by incorporating a special well balancing technique that is able to maintain particular stationary solutions of the governing PDE system up to machine precision. In particular, we consider steady vortex solutions of the shallow water equations, where the pressure gradient is in equilibrium with the centrifugal forces.
A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of the new method for both smooth and discontinuous problems. In particular we have compared the results for a steady vortex in equilibrium solved with a standard conforming ALE method (without any rezoning technique) and with our new nonconforming ALE scheme, to show that the new nonconforming scheme is able to avoid mesh distortion in vortex flows even after very long simulation times.

The study focusses on the dispersion and diffusion characteristics of discontinuous spectral element methods - specifically discontinuous Galerkin (DG) - via the spatial eigensolution analysis framework built around a one-dimensional linear problem, namely the linear advection equation. Dispersion and diffusion characteristics are of critical importance when dealing with under-resolved computations, as they affect both the numerical stability of the simulation and the solution accuracy. The spatial eigensolution analysis carried out in this paper complements previous analyses based on the temporal approach, which are more commonly found in the literature. While the latter assumes periodic boundary conditions, the spatial approach assumes inflow/outflow type boundary conditions and is therefore better suited for the investigation of open flows typical of aerodynamic problems, including transitional and fully turbulent flows and aeroacoustics. The influence of spurious/reflected eigenmodes is assessed with regard to the presence of upwind dissipation, naturally present in DG methods. This provides insights into the accuracy and robustness of these schemes for under-resolved computations, including under-resolved direct numerical simulation (uDNS) and implicit large-eddy simulation (iLES). The results estimated from the spatial eigensolution analysis are verified using the one-dimensional linear advection equation and successively by performing two-dimensional compressible Euler simulations that mimic (spatially developing) grid turbulence.

Hot spots ignition and shock to detonation transition modeling in pressed explosives is addressed in the frame of multiphase flow theory. Shock propagation results in mechanical disequilibrium effects between the condensed phase and the gas trapped in pores. Resulting subscale motion creates hot spots at pore scales. Pore collapse is modeled as a pressure relaxation process, during which dissipated power by the ‘configuration’ pressure produces local heating. Such an approach reduces 3D micromechanics and subscale contacts effects to a ‘granular’ equation of state. Hot spots criticity then results of the competition between heat deposition and conductive losses. Heat losses between the hot solid-gas interface at pore's scale and the colder solid core grains are determined through a subgrid model using two energy equations for the solid phase. The conventional energy balance equation provides the volume average solid temperature and a non-conventional energy equation provides the solid core temperature that accounts for shock heating. With the help of these two temperatures and subscale reconstruction, the interface temperature is determined as well as interfacial heat loss. The overall flow model thus combines a full disequilibrium two-phase model for the mean solid-gas flow variables with a subgrid model, aimed to compute local solid-gas interface temperature. Its evolution results of both subscale motion dissipation and conductive heat loss. The interface temperature serves as ignition criterion for the solid material deflagration. There is no subscale mesh, no system of partial differential equations solved at grain scale. The resulting model contains less parameter than existing ones and associates physical meaning to each of them. It is validated against experiments in two very different regimes: Shock to detonation transition, that typically happens in pressure ranges of 50 kbar and shock propagation that involves pressure ranges 10 times higher.

Atherosclerosis is an inflammatory disease due to the accumulation of low-density lipoproteins (LDLs) in the arteries wall, with the consequence that plaque is built up inside the arteries. Different mathematical models have been proposed to represent the first stages of atherosclerosis development. In this work we propose a mathematical model based on [A Hidalgo L Tello EF Toro. Numerical and analytical study of an atherosclerosis inflammatory disease model. Journal of Mathematical Biology 2014;68(7):1785–1814. 10.1007/s00285-013-0688-0],[N El Khatib S Genieys B Kazmierczak V Volpert. Reaction-diffusion model of atherosclerosis development. Journal of Mathematical Biology 2012;65(2):349–374. 10.1007/s00285-011-0461-1] but including a nonlinear diffusion porous medium-type in the 1D system of PDEs. The numerical solution is obtained by means of an ADER-WENO approach in the finite volume framework. First results theoretically obtained and numerically proved has been reported in [A Hidalgo L Tello EF Toro. Numerical and analytical study of an atherosclerosis inflammatory disease model. Journal of Mathematical Biology 2014;68(7):1785–1814. 10.1007/s00285-013-0688-0], for the linear diffusion model. In this work we extend those results to the nonlinear diffusion model.

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.

A general methodology, which consists in deriving two-dimensional finite-difference schemes which involve numerical fluxes based on Dirichlet-to-Neumann maps (or Steklov-Poincaré operators), is first recalled. Then, it is applied to several types of diffusion equations, some being weakly anisotropic, endowed with an external source. Standard finite-difference discretizations are systematically recovered, showing that in absence of any other mechanism, like e.g. convection and/or damping (which bring Bessel and/or Mathieu functions inside that type of numerical fluxes), these well-known schemes achieve a satisfying multi-dimensional character.

We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure. This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensional test-cases and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.

The sharp-interface resolution of compressible liquid–vapor flows is cumbersome due to the necessary tracking of the interface. The global dynamics are governed by local interface phenomena like surface tension effects as well as mass and energy transfer across the interface. These effects impose subtle jump conditions that have to be resolved locally at the interface. The exact solution of the two-phase Riemann problem is done by a complicated time-consuming iteration process. In addition most of the detailed knowledge of the Riemann pattern is not used in the overall algorithm such that the development of a much simpler approximative Riemann solver is desirable. The concept in this paper is to use an approximate solution of a two-phase Riemann problem at the phase interface. Extending the classical HLL Riemann solver for compressive shock waves, an additional intermediate state is introduced to distinguish between the liquid and vapor phase and to resolve the phase interface accurately. To get a thermodynamically consistent approximation a kinetic model is applied that determines the phase transition rate. The suitability of this approximative Riemann solver is validated by comparing the numerical results to the exact solution in one- and three-dimensional frameworks. The evaporation rates are compared to experiments with rapid evaporation and boiling. The two-phase approximative Riemann solver is then used in a compressible two-phase code with a sharp-interface resolution based on a ghost fluid approximation with data from this novel approximative two-phase Riemann solver. For a shock-droplet interaction this method provides a fast and accurate resolution of phase transfer and surface tension effects at much lower costs than the exact solution.

This work is dedicated to the development and comparison of WENO-type reconstructions for hyperbolic systems of balance laws. We are particularly interested in high order shock capturing non-oscillatory schemes with uniform accuracy within each cell and low spurious effects. We need therefore to develop a tool to measure the artifacts introduced by a numerical scheme. To this end, we study the deformation of a single Fourier mode and introduce the notion of distorsive errors, which measure the amplitude of the spurious modes created by a discrete derivative operator. Further we refine this notion with the idea of temperature, in which the amplitude of the spurious modes is weighted with its distance in frequency space from the exact mode. Following this approach linear schemes have zero temperature, but to prevent oscillations it is necessary to introduce nonlinearities in the scheme, thus increasing their temperature. However it is important to heat the linear scheme just enough to prevent spurious oscillations. With several tests we show that the newly introduced CWENOZ schemes are cooler than other existing WENO-type operators, while maintaining good non-oscillatory properties.

It is well known, thanks to Lax-Wendroff theorem, that the local conservation of a numerical scheme for a
conservative hyperbolic system is a simple and systematic way to guarantee that, if stable, a scheme will
provide a sequence of solutions that will converge to a weak solution of the continuous problem. In [1] it is
shown that a nonconservative scheme will not provide a good solution. The question of using, nevertheless, a
nonconservative formulation of the system and getting the correct solution has been a long-standing debate.
In this paper, we show how to get a relevant weak solution from a pressure-based formulation of the Euler
equations of fluid mechanics. This is useful when dealing with nonlinear equations of state because it is
easier to compute the internal energy from the pressure than the opposite. This makes it possible to get
oscillation-free solutions, contrarily to classical conservative methods. An extension to multiphase flows is
also discussed.

This paper is about numerical fluxes for hyperbolic systems and we first present a numerical flux, called GFORCE, that is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first-order upwind method. Then we incorporate GFORCE in the framework of the MUSTA approach [E.F. Toro, Multi-Stage Predictor–Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June, 2003], resulting in a version that we call GMUSTA. For non-linear systems this gives results that are comparable to those of the Godunov method in conjunction with the exact Riemann solver or complete approximate Riemann solvers, noting however that in our approach, the solution of the Riemann problem in the conventional sense is avoided. Both the GFORCE and GMUSTA fluxes are extended to multi-dimensional non-linear systems in a straightforward unsplit manner, resulting in linearly stable schemes that have the same stability regions as the straightforward multi-dimensional extension of Godunov’s method. The methods are applicable to general meshes. The schemes of this paper share with the family of centred methods the common properties of being simple and applicable to a large class of hyperbolic systems, but the schemes of this paper are distinctly more accurate. Finally, we proceed to the practical implementation of our numerical fluxes in the framework of high-order finite volume WENO methods for multi-dimensional non-linear hyperbolic systems. Numerical results are presented for the Euler equations and for the equations of magnetohydrodynamics.

We present a method for solving the generalized Riemann problem for partial differential equations of the advection–reactio type. The generalization of the Riemann problem here is twofold. Firstly, the governing equations include nonlinear advectio as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure local and valid for sufficiently small times, reduces the solution of the generalized Riemann problem of the inhomogeneou nonlinear equations to that of solving a sequence of conventional Riemann problems for homogeneous advection equations fo spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneou Burgers equation and the nonlinear shallow–water equations with variable bed elevation.

The generalized Riemann problem (GRP) is the initial value problem for a conservation law with piecewise smooth, but discontinuous initial data. We provide a new method for solving the GRP approximately, that can be used as a building block for high order finite volume or discontinuous Galerkin methods. Our new GRP solvers use the approximate states and wave speeds obtained through a HLL-type Riemann solver and use this information to build an approximation of the state in the GRP of any order. What is new about this approach compared to most previous solvers is that we no longer need to solve a classical Riemann problem exactly. We give a detailed explanation of this strategy for HLL and HLLC solvers for the Euler equations, as well as for the HLLD solver for MHD equations. We demonstrate the performance of the solvers from this new family of GRP solvers for a broad range of test problems.

The paper deals with the construction and analysis of efficient high order finite volume shock capturing schemes for the numerical solution of hyperbolic systems with stiff relaxation. In standard high order finite volume schemes it is difficult to treat the average of the source implicitly, since the computation of such average couples neighboring cells, making implicit schemes extremely expensive. The main novelty of the paper is that the average of the source is split into the sum of the source evaluated at the cell average plus a correction term. The first term is treated implicitly, while the small correction is treated explicitly, using IMEX-Runge-Kutta methods, thus resulting in a very effective semi-implicit scheme. This approach allows the construction of effective high order schemes in space and time. An asymptotic analysis is performed for small values of the relaxation parameter, giving an indication on the structure of the IMEX schemes that have to be adopted for time discretization. Several numerical tests confirm the accuracy and efficiency of the approach.

In this paper we analyze the use of time splitting techniques for solving shallow water equations. We discuss some properties that these schemes should satisfy so that interactions between the source term and the shock waves are controlled. This work shows that these schemes must be well balanced in the meaning expressed by Greenberg and Leroux [7]. More specifically, we analyze in what cases it is enough to verify an Approximate C-property and in which cases it is required to verify an Exact C-property (see [1, 2]). We also discuss this technique in two dimensions and include some numerical tests in order to justify our argument.

A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge–Kutta schemes. Compared to Runge–Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansion based on exact derivatives. The numerical results that are obtained with our proposal are satisfactory and show that this approximate approach can attain results as good as the exact Taylor procedure with less implementation and computational effort.

Several dispersion relation-preserving (DRP) spatially central discretizations are considered as the base scheme in the framework of the Yee & Sjögreen low dissipative nonlinear filter approach. In addition, the nonlinear filter of Yee & Sjögreen with shock-capturing and long time integration capabilities is used to replace the standard DRP linear filter for both smooth flows and flows containing discontinuities. DRP schemes for computational aeroacoustics (CAA) focus on dispersion error consideration for long time linear wave propagation rather than the formal order of accuracy of the scheme. The resulting DRP schemes usually have wider grid stencils and increased CPU operations count compared with standard central schemes of the same formal order of accuracy. For discontinuous initial data and long time wave propagation of smooth acoustic waves, various space and time DRP linear filter are needed. For acoustic waves interacting with shocks and turbulence induced noise, DRP schemes with linear filters alone usually are not capable of simulating such flows. The investigation presented in this paper is focused on the possible gain in efficiency and accuracy by spatial DRP schemes over standard central schemes having the same grid stencil width for general direct numerical simulations (DNS) and large eddy simulations (LES) of compressible flows. Representative test cases for both smooth flows and problems containing discontinuities for 3D DNS of compressible gas dynamics are included.

In this work we present and compare three Riemann solvers for the artificial compressibility perturbation of the 1D variable density incompressible Euler equations. The goal is to devise an artificial compressibility flux formulation to be used in Finite Volume or discontinuous Galerkin discretizations of the variable density incompressible Navier-Stokes equations. Starting from the constant density case, two Riemann solvers taking into account density jumps at fluid interfaces are first proposed. By enforcing the divergence free constraint in the continuity equation, these approximate Riemann solvers deal with density as a purely advected quantity. Secondly, by retaining the conservative form of the continuity equation, the exact Riemann solver is derived. The variable density solution is fully coupled with velocity and pressure unknowns. The Riemann solvers are compared and analysed in terms of robustness on harsh 1D Riemann problems. The extension to multidimensional problems is described. The effectiveness of the exact Riemann solver is demonstrated in the context of an high-order accurate discontinuous Galerkin discretization of variable density incompressible flow problems. We numerically validate the implementation considering the Kovasznay test case and the Rayleigh-Taylor instability problem.

We develop the Edge-Based Reconstruction WENO scheme for solving Euler equations on unstructured meshes. It belongs to the class of edge-based schemes with quasi-1D reconstruction of variables. The scheme monotonization is provided by using a convex combination of three lower-order reconstructions of variables in a similar way as it is in the classical finite-difference WENO scheme. The new scheme damps oscillations near shocks on unstructured meshes and, due to its edge-based nature, requires rather low computational costs. The properties of the new scheme are demonstrated on several test problems.

The goal of this work is to obtain a family of explicit high order well-balanced methods for the shallow water equations in spherical coordinates. Application of shallow water models to large scale problems requires the use of spherical coordinates: this is the case, for instance, of the simulation of the propagation of a Tsunami wave through the ocean. Although the PDE system is similar to the shallow water equations in Cartesian coordinates, new source terms appear. As a consequence, the derivation of high order numerical methods that preserve water at rest solutions is not as straightforward as in that case. Finite Volume methods are considered based on a first order path-conservative scheme and high order reconstruction operators. Numerical methods based on these ingredients have been successfully applied previously to the nonlinear SWEs in Cartesian coordinates. Some numerical tests to check the well-balancing and high order properties of the scheme, as well as its ability to simulate planetary waves or tsunami waves over realistic bathymetries are presented.

In this article we show the gain in accuracy and robustness brought by the use of a a posteriori MOOD limiting in replacement of the classical slope limiter employed in the remap phase of a legacy second-order Lagrange+Remap scheme solving the Euler system of equations. This simple substitution ensures extended robustness property, better accuracy and ability to capture physical phenomena. Numerical tests in 2D assess those improvements and the relative low cost of this a posteriori approach by reporting the number of troubled cells which demand re-computation. Situations like the occurrence of Not-a-Number, negative density and spurious numerical oscillations can therefore be cured.

In the active sound control problem a bounded domain is protected from noise generated outside via implementation of secondary sound sources on the perimeter. In the current paper we consider a quite general formulation in which sound sources are allowed to exist in the region to be shielded. The sound generated by the interior sources is considered as desired. It is required to remain it unaffected by the control in the protected area. This task proves to be much more complicated than the standard problem of active sound control because of the reverse effect of the controls on the input data. A novel practical algorithm is proposed that can be used for a real-time control. It accepts a preliminary tuning of the control system. In the algorithm the only input information eventually needed is the total acoustic field near the perimeter of the region to be shielded. It includes the contribution from both primary and secondary sources. In the algorithm the noise component to be attenuated is automatically extracted from the total acoustic field. The control system can potentially operate in a real-time regime since it only requires a consequent solution of a quadratic programming problem.

The new two-time-level dispersion improved CABARET scheme is developed as an upgrade of the original CABARET for improved wave propagation modelling in multiple dimensions and for nonlinear conservation laws including gas dynamics. The new upgrade retains many attractive features of the original CABARET scheme such as shock-capturing and low dissipation. It is simple for implementation in the existing CABARET codes and leads to a greater accuracy for solving linear wave propagation problems. A non-linear version of the dispersion-improved CABARET scheme is introduced to efficiently deal with contact discontinuities and shocks. The properties of the new linear and nonlinear CABARET schemes are analysed for numerical dissipation and dispersion error based on Von Neumann analysis and Pirrozolli's method. Numerical examples for one-dimensional and two-dimensional linear advection, the one-dimensional inviscid Burger's equation, and the isothermal gas dynamics problems in one and two dimensions are presented.

For a Lagrangian scheme solving the compressible Euler equations in cylindrical coordinates, two important issues are whether the scheme can maintain spherical symmetry (symmetry-preserving) and whether the scheme can maintain positivity of density and internal energy (positivity-preserving). While there were previous results in the literature either for symmetry-preserving in the cylindrical coordinates or for positivity-preserving in cartesian coordinates, the design of a Lagrangian scheme in cylindrical coordinates, which is high order in one-dimension and second order in two-dimensions, and can maintain both spherical symmetry-preservation and positivity-preservation simultaneously, is challenging. In this paper we design such a Lagrangian scheme and provide numerical results to demonstrate its good behavior.

In this paper we will present an existence and uniqueness result for an entropy solution of weakly coupled systems of conservation laws on moving surfaces without boundary. The evolution of the hypersurface is prescribed. The coupling of the system is realized by a source term not depending on derivatives of the unknown function.

This paper investigates the numerical accuracy of implicit Large Eddy Simulations (iLES) in relation to compressible turbulent boundary layers (TBL). iLES are conducted in conjunction with Monotonic Upstream-Centred Scheme for Conservation Laws (MUSCL) and Weighted Essentially Non-Oscillatory (WENO), ranging from 2nd to 9th-order. The accuracy effects are presented from a physical perspective showing skewness, flatness and anisotropy calculations, among others. The order of the scheme directly affects the physical representation of the TBL, especially the degree of asymmetry and anisotropy in the sub-layers of the TBL. The study concludes that high-order iLES can provide an accurate and detailed description of TBL directly comparable to available DNS and experimental results.

A first-order well-balanced finite volume scheme for the solution of a multi-component gas flow model in a pipe on non-flat topography is introduced. The mathematical model consists of Euler equations with source terms which arise from heat exchange, and gravity and viscosity forces, coupled with the mass conservation equations of species. We propose a segregated scheme in which the Euler and species equations are solved separately. This methodology leads to a flux vector in the Euler equations which depends not only on the conservative variables but also on time and space variables through the gas composition. This fact makes necessary to add some artificial viscosity to the usual numerical flux which is done by introducing an additional source term. Besides, in order to preserve the positivity of the species concentrations, we discretize the flux in the mass conservation equations for species, in accordance with the upwind discretization of the total mass conservation equation in the Euler system. Moreover, as proposed in a previous reference by the authors, [5], the discretizations of the flux and source terms are made so as to ensure that the full scheme is well-balanced. Numerical tests including both academic and real gas network problems are solved, showing the performance of the proposed methodology.

The paper is devoted to the further development of the numerical methods to solve model kinetic equations and their application to hypersonic rarefied gas flows. Firstly, we verify the accuracy of the approach by comparing our results with the well resolved DMSC calculations for argon and nitrogen. Secondly, computation of an external flow over a model winged re-entry vehicle with orbital free-stream velocity and 100 km altitude is shown. Influence of the spatial mesh type is studied. Finally, scalability of the implicit kinetic solver is demonstrated for up to 256 nodes.

In this paper we introduce a definition of the local conservation property for numerical methods solving time dependent conservation laws, which generalizes the classical local conservation definition. The motivation of our definition is the Lax-Wendroff theorem, and thus we prove it for locally conservative numerical schemes per our definition in one and two space dimensions. Several numerical methods, including continuous Galerkin methods and compact schemes, which do not fit the classical local conservation definition, are given as examples of locally conservative methods under our generalized definition.

We present a global, closed-loop, multiscale mathematical model for the human circulation including the arterial system, the venous system, the heart, the pulmonary circulation and the microcirculation. A distinctive feature of our model is the detailed description of the venous system, particularly for intracranial and extracranial veins. Medium to large vessels are described by one-dimensional hyperbolic systems while the rest of the components are described by zero-dimensional models represented by differential-algebraic equations. Robust, high-order accurate numerical methodology is implemented for solving the hyperbolic equations, which are adopted from a recent reformulation that includes variable material properties. Because of the large intersubject variability of the venous system, we perform a patient-specific characterization of major veins of the head and neck using MRI data. Computational results are carefully validated using published data for the arterial system and most regions of the venous system. For head and neck veins, validation is carried out through a detailed comparison of simulation results against patient-specific phase-contrast MRI flow quantification data. A merit of our model is its global, closed-loop character; the imposition of highly artificial boundary conditions is avoided. Applications in mind include a vast range of medical conditions. Of particular interest is the study of some neurodegenerative diseases, whose venous haemodynamic connection has recently been identified by medical researchers. Copyright © 2014 John Wiley & Sons, Ltd.

We present a well-balanced, high-order non-linear numerical scheme for solving a hyperbolic system that models one-dimensional flow in blood vessels with variable mechanical and geometrical properties along their length. Using a suitable set of test problems with exact solution, we rigorously assess the performance of the scheme. In particular, we assess the well-balanced property and the effective order of accuracy through an empirical convergence rate study. Schemes of up to fifth order of accuracy in both space and time are implemented and assessed. The numerical methodology is then extended to realistic networks of elastic vessels and is validated against published state-of-the-art numerical solutions and experimental measurements. It is envisaged that the present scheme will constitute the building block for a closed, global model for the human circulation system involving arteries, veins, capillaries and cerebrospinal fluid. Copyright © 2013 John Wiley & Sons, Ltd.

Compressible multi-phase flows are found in a variety of scientific and engineering problems. The development of accurate and efficient numerical algorithms for multi-phase flow simulations remains one of the challenging issues in computational fluid dynamics. A main difficulty of numerical methods for multi-phase flows is that the model equations cannot always be written in conservative form, though they may be hyperbolic and derived from physical conservation principles. In this work, assuming a hyperbolic model, a path-conservative method is developed to deal with the non-conservative character of the equations. The method is applied to solve the five-equation model of Saurel and Abgrall for two-phase flow. As another contribution of the work, a simplified HLLC-type approximate Riemann solver is proposed to compute the Godunov state to be incorporated into the Godunov-type path-conservative method. A second order, semi-discrete version of the method is then constructed via a MUSCL reconstruction with Runge–Kutta time stepping. Moreover, the method is then extended to the two-dimensional case by directional splitting. The method is systematically assessed via a series of test problems with exact solutions, finding satisfactory results.

New first- and high-order centred methods for conservation laws are presented. Convenient TVD conditions for constructing centred TVD schemes are then formulated and some useful results are proved. Two families of centred TVD schemes are constructed and extended to nonlinear systems. Some numerical results are also presented.

Experimental and numerical results concerning the flow induced by the break of a dam on a dry bed are presented. The numerical technique consists of a shock-capturing method of the Godunov type. A physical laboratory model has been employed to infer properties and validity of the numerical solution. Attention is also given to the applicability of the mathematical model, based on the shallow water equations, to this class of problems.

We explore how the weighted average flux approach can be used to generate first- and second-order accurate finite volume schemes for the linear advection equatons in one, two, and three space dimensions. The derived schemes have multidimensional upwinding aspects and good stability properties. From the two-dimensional methods, we construct a scheme for nonlinear systems of hyperbolic conservation laws that is second-order accurate in smooth flow. Spurious oscillations are controlled by making use of one-dimensional TVD limiter functions. Numerical results are presented for the shallow water equations in two space dimensions. The equivalent schemes are derived for nonlinear systems in three space dimensions.

We present first- and higher-order non-oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non-conservative) form. Non-conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non-conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second-order non-oscillatory methods. We have used the MUSCL-Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher-order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non-conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind-based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd.

The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.

The equations of hydrodynamics are modified by the inclusion of additional terms which greatly simplify the procedures needed for stepwise numerical solution of the equations in problems involving shocks. The quantitative influence of these terms can be made as small as one wishes by choice of a sufficiently fine mesh for the numerical integrations. A set of difference equations suitable for the numerical work is given, and the condition that must be satisfied to insure their stabilty is derived.

We first construct an approximate Riemann solver of the HLLC-type for the Baer–Nunziato equations of compressible two-phase flow for the “subsonic” wave configuration. The solver is fully nonlinear. It is also complete, that is, it contains all the characteristic fields present in the exact solution of the Riemann problem. In particular, stationary contact waves are resolved exactly. We then implement and test a new upwind variant of the path-conservative approach; such schemes are suitable for solving numerically nonconservative systems. Finally, we use locally the new HLLC solver for the Baer–Nunziato equations in the framework of finite volume, discontinuous Galerkin finite element and path-conservative schemes. We systematically assess the solver on a series of carefully chosen test problems.

In this paper we present the exact solution of the Riemann problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave. We provide examples of possible wave patterns. Numerical solution of a first-order dissipative scheme as well as an implementation of our Riemann solver in the second-order upwind method are compared with the proposed exact Riemann problem solution. A practical implementation of the proposed exact Riemann solver in the framework of a second-order upwind TVD method is also illustrated.

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.

In this paper, we first briefly review the semi-analytical method [E.F. Toro, V.A. Titarev, Solution of the generalized Riemann problem for advection–reaction equations, Proc. Roy. Soc. London 458 (2018) (2002) 271–281] for solving the derivative Riemann problem for systems of hyperbolic conservation laws with source terms. Next, we generalize it to hyperbolic systems for which the Riemann problem solution is not available. As an application example we implement the new derivative Riemann solver in the high-order finite-volume ADER advection schemes. We provide numerical examples for the compressible Euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes.

A unified, Riemann-problem-based extension of the Warming–Beam and Lax- Wendroff schemes for nonlinear systems of hyperbolic conservation laws is presented. This is achieved by employing the weighted average flux (WAF) approach with the numerical flux given by a space–time integral average of the solution of local Riemann problems. The local wave structure dictates switching between schemes automatically with no need for special conservative switching operators. The resulting Godunov-type scheme has CFL-number 2 stability restriction and when used in conjunction with Riemann-solver adaption procedures is significantly more efficient than Riemann-problem-based methods currently in use.