ADER Schemes Research Articles

Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables.

We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. To obtain a conservative method, the underlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are subsequently converted into point values of the primitive variables. This is the only place where the conversion from conservative to primitive variables is needed in the new scheme. Then, a second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. The resulting space-time polynomials of the primitive variables can then be directly used as input for the numerical fluxes at the cell boundaries in the underlying conservative finite volume scheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced to just one single conversion at each cell center. We have verified the validity of the new approach over a wide range of hyperbolic systems, including the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and the CPU time is reduced by about 25 %. Because of its increased accuracy and due to the reduced computational cost, we recommend to use this version of ADER as the standard one in the relativistic framework. At the end of the paper, the new approach has also been extended to ADER-DG schemes on space-time adaptive grids (AMR).

Finite difference elastic wave modeling with an irregular free surface using ADER scheme

In numerical modeling of seismic wave propagation in the earth, we encounter two important issues: the free surface and the topography of the surface (i.e. irregularities). In this study, we develop a 2D finite difference solver for the elastic wave equation that combines a 4th- order ADER scheme (Arbitrary high-order accuracy using DERivatives), which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. The idea is to treat the free surface boundary explicitly by using ghost values of the solution for points beyond the free surface to impose the physical boundary condition. The method is based on the velocity-stress formulation. The ultimate goal is to develop a numerical solver for the elastic wave equation that is stable, accurate and computationally efficient. The solver treats smooth arbitrary-shaped boundaries as simple plane boundaries. The computational cost added by treating the topography is negligible compared to flat free surface because only a small number of grid points near the boundary need to be computed. In the presence of topography, using 10 grid points per shortest shear-wavelength, the solver yields accurate results. Benchmark numerical tests using several complex models that are solved by our method and other independent accurate methods show an excellent agreement, confirming the validity of the method for modeling elastic waves with an irregular free surface.

Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations

In this work, an ADER type finite volume numerical scheme is proposed as an extension of a first order solver based on weak solutions of RPs with source terms. The type of source terms considered here are a special but relevant type of source terms: their spatial integral is discontinuous. The relevant difference with other previously defined ADER schemes is that it considers the presence of the source term in the solutions of the DRP. Unlike the original ADER schemes, the proposed numerical scheme computes the RPs of the high order terms of the DRP departing from time derivatives of the fluxes as initial conditions for these RPs. Weak solutions of the RPs defined for the DRP are computed using an augmented version of the Roe solver that includes an extra wave that accounts for the contribution of the source term. The discretization done over the source term leads to an energy balanced numerical scheme that allows to obtain the exact solution for steady cases with independence of the grid refinement. In unsteady problems, the numerical scheme ensures the convergence to the exact solution. The numerical scheme is constructed with an arbitrary order of accuracy, and has no theoretical barrier. Numerical results for the Burger's equation and the shallow water equations are presented in this work and indicate that the proposed numerical scheme is able to converge with the expected order of accuracy.

Wave simulation in 2D heterogeneous transversely isotropic porous media with fractional attenuation: A Cartesian grid approach

A time-domain numerical modeling of transversely isotropic Biot poroelastic waves is proposed in two dimensions. The viscous dissipation occurring in the pores is described using the dynamic permeability model developed by Johnson–Koplik–Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency. In the time-domain, these coefficients introduce shifted fractional derivatives of order 1/2, involving a convolution product. Based on a diffusive representation, the convolution kernel is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations, resulting in the Biot-DA (diffusive approximation) model. The properties of both the Biot-JKD and the Biot-DA models are analyzed: hyperbolicity, decrease of energy, dispersion. To determine the coefficients of the diffusive approximation, two approaches are analyzed: Gaussian quadratures and optimization methods in the frequency range of interest. The nonlinear optimization is shown to be the better way of determination. A splitting strategy is then applied to approximate numerically the Biot-DA equations. The propagative part is discretized using a fourth-order ADER scheme on a Cartesian grid, whereas the diffusive part is solved exactly. An immersed interface method is implemented to take into account heterogeneous media on a Cartesian grid and to discretize the jump conditions at interfaces. Numerical experiments are presented. Comparisons with analytical solutions show the efficiency and the accuracy of the approach, and some numerical experiments are performed to investigate wave phenomena in complex media, such as multiple scattering across a set of random scatterers.

Reformulations for general advection–diffusion–reaction equations and locally implicit ADER schemes

Following Cattaneo's original idea, in this article we first present two relaxation formulations for time-dependent, non-linear systems of advection–diffusion–reaction equations. Such formulations yield time-dependent non-linear hyperbolic balance laws with stiff source terms. Then we present a locally implicit version of the ADER method to solve these stiff systems to high accuracy. The new ingredient of the numerical methodology is a locally implicit solution of the generalised Riemann problem. We illustrate the formulations and the resulting numerical approach by solving the compressible Navier–Stokes equations.

ADER schemes and high order coupling on networks of hyperbolic conservation laws

In this article we present a method to extend high order finite volume schemes to networks of hyperbolic conservation laws with algebraic coupling conditions. This method is based on an ADER approach in time to solve the generalized Riemann problem at the junction. Additionally to the high order accuracy, this approach maintains an exact conservation of quantities if stated by the coupling conditions. Several numerical examples confirm the benefits of a high order coupling procedure for high order accuracy and stable shock capturing.

A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme

We introduce a sub-cell WENO reconstruction method to evaluate spatial derivatives in the high-order ADER scheme. The basic idea in our reconstruction is to use only r stencils to reconstruct the point-wise values of solutions and spatial derivatives for the (2r-1)th-order ADER scheme in one dimension, while in two dimensions, the dimension-by-dimension sub-cell reconstruction approach for spatial derivatives is employed. Compared with the original ADER scheme of Toro and Titarev (2002) [2] that uses the direct derivatives of reconstructed polynomials for solutions to evaluate spatial derivatives, our method not only reduces greatly the computational costs of the ADER scheme on a given mesh, but also avoids possible numerical oscillations near discontinuities, as demonstrated by a number of one- and two-dimensional numerical tests. All these tests show that the 5th-order ADER scheme based on our sub-cell reconstruction method achieves the desired accuracy, and is essentially non-oscillatory and computationally cheaper for problems with discontinuities.

Wave Propagation Across Acoustic/Biot’s Media: A Finite-Difference Method

AbstractNumerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media. Wave propagation is described by the usual acoustics equations (in the fluid medium) and by the low-frequency Biot’s equations (in the porous medium). Interface conditions are introduced to model various hydraulic contacts between the two media: open pores, sealed pores, and imperfect pores. Well-posedness of the initial-boundary value problem is proven. Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context: a fourth-order ADER scheme with Strang splitting for time- marching; a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory; and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution. Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions, demonstrating the accuracy of the approach.

Biot-JKD model: Simulation of 1D transient poroelastic waves with fractional derivatives

A time-domain numerical modeling of Biot poroelastic waves is presented. The viscous dissipation occurring in the pores is described using the dynamic permeability model developed by Johnson–Koplik–Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency: in the time-domain, these coefficients introduce order 1/2 shifted fractional derivatives involving a convolution product. Based on a diffusive representation, the convolution kernel is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations. Thanks to the dispersion relation, the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest. A splitting strategy is then applied numerically: the propagative part of Biot-JKD equations is discretized using a fourth-order ADER scheme on a Cartesian grid, whereas the diffusive part is solved exactly. Comparisons with analytical solutions show the efficiency and the accuracy of this approach.

Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – Speed comparisons with Runge–Kutta methods

ADER (Arbitrary DERivative in space and time) methods for the time-evolution of hyperbolic conservation laws have recently generated a fair bit of interest. The ADER time update can be carried out in a single step, which is desirable in many applications. However, prior papers have focused on the theory while downplaying implementation details. The purpose of the present paper is to make ADER schemes accessible by providing two useful formulations of the method as well as their implementation details on three-dimensional structured meshes. We therefore provide a detailed formulation of ADER schemes for conservation laws with non-stiff source terms in nodal as well as modal space along with useful implementation-related details. A good implementation of ADER requires a fast method for transcribing from nodal to modal space and vice versa and we provide innovative transcription strategies that are computationally efficient. We also provide details for the efficient use of ADER schemes in obtaining the numerical flux for conservation laws as well as electric fields for divergence-free magnetohydrodynamics (MHD). An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magnetic fields in MHD is also presented. Several explicit formulae have been provided in all instances for ADER schemes spanning second to fourth orders.The schemes catalogued here have been implemented in the first author’s RIEMANN code. The speed of ADER schemes is shown to be almost twice as fast as that of strong stability preserving Runge–Kutta time stepping schemes for all the orders of accuracy that we tested. The modal and nodal ADER schemes have speeds that are within ten percent of each other. When a linearized Riemann solver is used, the third order ADER schemes are half as fast as the second order ADER schemes and the fourth order ADER schemes are a third as fast as the third order ADER schemes. The third order ADER scheme, either with an HLL or linearized Riemann solver, represents an excellent upgrade path for scientists and engineers who are working with a second order Runge–Kutta based total variation diminishing (TVD) scheme. Several stringent test problems have been catalogued. [Display omitted]

Numerical model of swash motion and air entrapment within coarse-grained beaches

The paper presents a numerical model for bore-driven swash on permeable coarse-grained beaches. The surface flow module is modelled using the non-linear shallow water equations (NLSWEs), solved using the Godunov-based finite volume ADER scheme, which is suitable for handling steep bores as well as large source terms in the NLSWEs. The subsurface flow comprises: (i) infiltration and exfiltration modelled as vertical piston-like flow, (ii) horizontal pore-air movement within the unsaturated region of the beach, (iii) the horizontal groundwater flow. Model predictions of the surface and subsurface flow are in good agreement with measurements from large-scale laboratory experiments for swash on permeable, immobile beaches. In these simulations air velocity was sufficiently small to justify using Darcy's resistance law for the air flow, whereas the quadratic Forchheimer law had to be used for the infiltration and groundwater flow. The validated numerical model provides insight into the surface-subsurface water exchange for bore-driven swash on coarse-grained beaches. The impact of air entrapped between the wetting front and the groundwater level on the water exchange is examined in particular.

Application of ADER Scheme in MHD Simulation

The Arbitrary accuracy Derivatives Riemann problem method (ADER) scheme is a new high order numerical scheme based on the concept of finite volume integration, and it is very easy to be extended up to any order of space and time accuracy by using a Taylor time expansion at the cell interface position. So far the approach has been applied successfully to flow mechanics problems. Our objective here is to carry out the extension of multidimensional ADER schemes to multidimensional MHD systems of conservation laws by calculating several MHD problems in one and two dimensions: (ⅰ) Brio-Wu shock tube problem, (ⅱ) Dai-Woodward shock tube problem, (ⅲ) Orszag-Tang MHD vortex problem. The numerical results prove that the ADER scheme possesses the ability to solve MHD problem, remains high order accuracy both in space and time, keeps precise in capturing the shock. Meanwhile, the compared tests show that the ADER scheme can restrain the oscillation and obtain the high order non-oscillatory result.

Numerical modeling of transient two-dimensional viscoelastic waves

This paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids. Dissipation mechanisms are described using the generalized Zener model. No time convolutions are required thanks to the introduction of memory variables that satisfy local-in-time differential equations. By appropriately choosing the relaxation parameters, it is possible to accurately describe a large range of materials, such as solids with constant quality factors. The evolution equations satisfied by the velocity, the stress, and the memory variables are written in the form of a first-order system of PDEs with a source term. This system is solved by splitting it into two parts: the propagative part is discretized explicitly, using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is then solved exactly. Jump conditions along the interfaces are discretized by applying an immersed interface method. Numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems, such as multiple scattering configurations.

Time domain numerical modeling of wave propagation in 2D heterogeneous porous media

This paper deals with the numerical modeling of wave propagation in porous media described by Biot’s theory. The viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity, which is valid in the low-frequency range. The coexistence of propagating fast compressional wave and shear wave, and of a diffusive slow compressional wave, makes numerical modeling tricky. To avoid restrictions on the time step, the Biot’s system is splitted into two parts: the propagative part is discretized by a fourth-order ADER scheme, while the diffusive part is solved analytically. Near the material interfaces, a space–time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave. The jump conditions along the interfaces are discretized by an immersed interface method. Numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling. The efficiency of the approach is illustrated by simulations of multiple scattering.

ADER Schemes for Nonlinear Systems of Stiff Advection–Diffusion–Reaction Equations

In this article we extend the high order ADER finite volume schemes introduced for stiff hyperbolic balance laws by Dumbser, Enaux and Toro (J. Comput. Phys. 227:3971---4001, 2008) to nonlinear systems of advection---diffusion---reaction equations with stiff algebraic source terms. We derive a new efficient formulation of the local space-time discontinuous Galerkin predictor using a nodal approach whose interpolation points are tensor-products of Gauss---Legendre quadrature points. Furthermore, we propose a new simple and efficient strategy to compute the initial guess of the locally implicit space-time DG scheme: the Gauss---Legendre points are initialized sequentially in time by a second order accurate MUSCL-type approach for the flux term combined with a Crank---Nicholson method for the stiff source terms. We provide numerical evidence that when starting with this initial guess, the final iterative scheme for the solution of the nonlinear algebraic equations of the local space-time DG predictor method becomes more efficient. We apply our new numerical method to some systems of advection---diffusion---reaction equations with particular emphasis on the asymptotic preserving property for linear model systems and the compressible Navier---Stokes equations with chemical reactions.

Numerical modeling of 1D transient poroelastic waves in the low-frequency range

Propagation of transient mechanical waves in porous media is numerically investigated in 1D. The framework is the linear Biot model with frequency-independent coefficients. The coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky. A method combining three numerical tools is proposed: a fourth-order ADER scheme with time-splitting to deal with the time-marching, a space–time mesh refinement to account for the small-scale evolution of the slow wave, and an interface method to enforce the jump conditions at interfaces. Comparisons with analytical solutions confirm the validity of this approach.

ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P N P M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P N P M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].

Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax–Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.

Numerical modeling of transient poroelastic waves in the low frequency range

A numerical method is proposed to simulate the propagation of transient poroelastic waves across heterogeneous media, in the low frequency range. A velocity-stress formulation of Biot's equations is followed, leading to a first-order differential system. The latter is splitted in two parts: a propagative one discretized by a fourth-order ADER scheme, and a diffusive one solved analytically. Near sources and interfaces, a space-time mesh refinement is implemented to capture the small scales of evolution of the diffusive slow compressional wave. Lastly, an immersed interface method is implemented to accurately model the jump conditions at interfaces between the different media. Numerical experiments in one and two dimensions are shown, with porous/porous or fluid/porous interfaces. Comparisons with analytical solutions confirm the efficiency of the approach. [1] G. Chiavassa, B. Lombard, J. Piraux, Numerical modeling of 1-D transient poroelastic waves in the low-frequency range, soumis au J. Comput. Appl. Math., (2007), disponible sur http://hal.archives-ouvertes.fr/hal-00193103/fr/

Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws

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.