High resolution upwind and centred methods are today a mature generation of computational techniques applicable to a wide range of engineering and scientific disciplines, Computational Fluid Dynamics ...(CFD) being the most prominent up to now. This textbook gives a comprehensive, coherent and practical presentation of this class of techniques. The book is designed to provide readers with an understanding of the basic concepts, some of the underlying theory, the ability to critically use the current research papers on the subject, and, above all, with the required information for the practical implementation of the methods. Direct applicability of the methods include: compressible, steady, unsteady, reactive, viscous, non-viscous and free surface flows. For this third edition the book was thoroughly revised and contains substantially more, and new material both in its fundamental as well as in its applied parts.
In this article, a conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree
N are used as ...test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree
M
⩾
N
, which are reconstructed from the underlying polynomials of degree
N. This approach yields a general, unified framework that contains as two special cases classical high order finite volume (FV) schemes
(
N
=
0
)
as well as the usual discontinuous Galerkin (DG) method
(
N
=
M
)
. In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing
N
≠
0
and
M
>
N
. The reconstruction operator is implemented for arbitrary polynomial degrees
N and
M on unstructured triangular and tetrahedral meshes in two and three space dimensions.
To provide a high order accurate one-step time integration of the same formal order of accuracy as the spatial discretization operator, the (reconstructed) polynomial data of degree
M are evolved in time locally inside each element using a new
local continuous space–time Galerkin method. As a result of this approach, we obtain, as a high order accurate predictor, space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms, which then can be used in a natural way to construct very efficient
fully-discrete and
quadrature-free one-step schemes. This feature is particularly important for DG schemes in three space dimensions, where the cost of numerical quadrature may become prohibitively expensive for very high orders of accuracy.
Numerical convergence studies of all members of the new general class of proposed schemes are shown up to sixth-order of accuracy in space
and time on unstructured two- and three-dimensional meshes for two very prominent nonlinear hyperbolic systems, namely for the Euler equations of compressible gas dynamics and the equations of ideal magnetohydrodynamics (MHD). The results indicate that the new class of intermediate schemes
(
N
≠
0
,
M
>
N
)
is computationally more efficient than classical finite volume or DG schemes.
Finally, a large set of interesting test cases is solved on unstructured meshes, where the proposed new time stepping approach is applied to the equations of ideal and relativistic MHD as well as to nonlinear elasticity, using a standard high order WENO finite volume discretization in space to cope with discontinuous solutions.
In this article, we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with
stiff source terms. The new class of schemes is ...based on a three stage procedure. First a high-order WENO reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed
locally inside each cell using the governing equations. In the original ENO scheme of Harten et al. and in the ADER schemes of Titarev and Toro, this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time, i.e. by applying the so-called Cauchy–Kovalewski procedure. However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy–Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a
local space–time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space–time integration over each control volume, using the local space–time DG solutions at the Gaussian integration points for the intercell fluxes and for the space–time integral over the source term. We will show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of LeVeque and Yee R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of Computational Physics 86 (1) (1990) 187–210, the relaxation system of Jin and Xin S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics 48 (1995) 235–277 and the full compressible Euler equations with stiff friction source terms.
We propose a simple extension of the well-known Riemann solver of Osher and Solomon (Math. Comput. 38:339–374,
1982
) to a certain class of hyperbolic systems in non-conservative form, in particular ...to shallow-water-type and multi-phase flow models. To this end we apply the formalism of path-conservative schemes introduced by Parés (SIAM J. Numer. Anal. 44:300–321,
2006
) and Castro et al. (Math. Comput. 75:1103–1134,
2006
). For the sake of generality and simplicity, we suggest to compute the inherent path integral numerically using a Gaussian quadrature rule of sufficient accuracy. Published path-conservative schemes to date are based on either the Roe upwind method or on centered approaches. In comparison to these, the proposed new path-conservative Osher-type scheme has several advantages. First, it does not need an entropy fix, in contrast to Roe-type path-conservative schemes. Second, our proposed non-conservative Osher scheme is very simple to implement and nonetheless constitutes a
complete
Riemann solver in the sense that it attributes a different numerical viscosity to each characteristic field present in the relevant Riemann problem; this is in contrast to centered methods or
incomplete
Riemann solvers that usually neglect intermediate characteristic fields, hence leading to excessive numerical diffusion. Finally, the interface jump term is differentiable with respect to its arguments, which is useful for steady-state computations in implicit schemes. We also indicate how to extend the method to general unstructured meshes in multiple space dimensions. We show applications of the first order version of the proposed path-conservative Osher-type scheme to the shallow water equations with variable bottom topography and to the two-fluid debris flow model of Pitman & Le. Then, we apply the higher-order multi-dimensional version of the method to the Baer–Nunziato model of compressible multi-phase flow. We also clearly emphasize the
limitations
of our approach in a special chapter at the end of this article.
The importance of the study of the blood flow equations is widely recognized as it is a tool to understand the circulatory system. Arteries and veins result to have both elastic and viscous ...behaviour. Models for the first case are much more studied as they result to be simpler and still satisfying if compared to experimental data. In this paper, we consider a model which encompasses both the elastic and viscoelastic response in arterial walls, respectively leading to a conservative and a non-conservative system. We present a second-order scheme based on the first-order Price-T scheme and the MUSCL-Hancock strategy. This approach automatically adapts to the above conservative and non-conservative cases.
Then, we perform a Sensitivity Analysis (SA) based on the Continuous Sensitivity Equation Method (CSEM), whose aim is the study of how changes in the inputs of a model can affect its outputs. In particular, the sensitivity is defined as the derivative (with respect to an uncertain parameter a) of the solution of the system taken into consideration. Since the CSEM cannot be directly applied to discontinuous solutions, we add a source term to compensate the spikes associated to the Dirac delta functions that can arise in the sensitivity variables.
One of the main applications of SA is uncertainty quantification, which is investigated for a Riemann problem as well as for a network of 37 arteries. Details on junctions for coupling two or more vessels are also given.
•Non-conservative MUSCL-Hancock method that naturally reduces to a conservative scheme.•Continuous sensitivity equation method applied to viscoelastic blood flow model.•Uncertainty quantification with respect to the arterial stiffness.•Application to a human arterial network of 37 vessels; linearized coupling condition.
•Non-trivial moving steady state solutions because of the presence of the friction source term.•Well-balanced strategy needed to get correct approximation of source terms.•New well-balanced approach ...based on a Godunov-type scheme.•Steady states exactly preserved by the fully well-balanced scheme.•Positivity of the vessel cross-sectional area ensured by the scheme.
We consider the 1D blood flow equations with friction source term. The main purpose of this work is the derivation of a positivity preserving and fully well-balanced scheme, which correctly approximates the solutions of this system, exactly preserves all the associated steady states, including the moving ones, and ensures the positivity of the cross-sectional area. To address such issues, a study of the moving steady states related to the friction source term is first performed. Afterwards, a Godunov-type scheme is constructed, by deriving a two-state approximate Riemann solver and by introducing a relevant discretization of the source term, to enforce the well-balanced property. The scheme is then adapted to the generalized model including a space-variable viscous resistance; a second-order well-balanced MUSCL extension of the scheme is also proposed. Numerical experiments are finally carried out in order to validate the scheme and highlight its properties.
We propose a new robust and accurate SPH scheme, able to track correctly complex three-dimensional non-hydrostatic free surface flows and, even more important, also able to compute an accurate and ...little oscillatory pressure field. It uses the explicit third order TVD Runge–Kutta scheme in time, following Shu and Osher Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys 1988;89:439–71, together with the new key idea of introducing a monotone upwind flux for the density equation, thus removing any artificial viscosity term. For the discretization of the velocity equation, the non-diffusive central flux has been used. A new flexible approach to impose the boundary conditions at solid walls is also proposed. It can handle any moving rigid body with arbitrarily irregular geometry. It does neither produce oscillations in the fluid pressure in proximity of the interfaces, nor does it have a restrictive impact on the stability condition of the explicit time stepping method, unlike the repellent boundary forces of Monaghan Monaghan JJ. Simulating free surface flows with SPH. J Comput Phys 1994;110:399–406. To asses the accuracy of the new SPH scheme, a 3D mesh-convergence study is performed for the strongly deforming free surface in a 3D dam-break and impact-wave test problem providing very good results.
Moreover, the parallelization of the new 3D SPH scheme has been carried out using the message passing interface (MPI) standard, together with a dynamic load balancing strategy to improve the computational efficiency of the scheme. Thus, simulations involving millions of particles can be run on modern massively parallel supercomputers, obtaining a very good performance, as confirmed by a speed-up analysis. The 3D applications consist of environmental flow problems, such as dam-break flows and impact flows against a wall. The numerical solutions obtained with our new 3D SPH code have been compared with either experimental results or with other numerical reference solutions, obtaining in all cases a very satisfactory agreement.
In this article we present a quadrature-free essentially non-oscillatory finite volume scheme of arbitrary high order of accuracy both in space and time for solving nonlinear hyperbolic systems on ...unstructured meshes in two and three space dimensions. For high order spatial discretization, a WENO reconstruction technique provides the reconstruction polynomials in terms of a hierarchical orthogonal polynomial basis over a reference element. The Cauchy–Kovalewski procedure applied to the reconstructed data yields for each element a space–time Taylor series for the evolution of the state and the physical fluxes. This Taylor series is then inserted into a special numerical flux across the element interfaces and is subsequently integrated analytically in space and time. Thus, the Cauchy–Kovalewski procedure provides a natural, direct and cost-efficient way to obtain a quadrature-free formulation, avoiding the expensive numerical quadrature arising usually for high order finite volume schemes in three space dimensions. We show numerical convergence results up to sixth order of accuracy in space and time for the compressible Euler equations on triangular and tetrahedral meshes in two and three space dimensions. Furthermore, various two- and three-dimensional test problems with smooth and discontinuous solutions are computed to validate the approach and to underline the non-oscillatory shock-capturing properties of the method.
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.
