High-Order ADER Numerical Methods and Applications to Science and Engineering

s of presentations E. F. Toro Mathematical modelling, numerical simulation and high-order ADER schemes In this lecture I review some basic concepts regarding the mathematical modelling of processes in physics and other disciplines and their numerical simulation. There follows a discussion on the issue of high-order of accuracy of numerical methods, its misconceptions and eventual justification. I then give a simple review of the ADER approach (see for example [1], [2]) for constructing non-linear numerical methods of arbitrary order of accuracy in space and time for solving partial differential equations that govern wave propagation phenomena. The corner stone of the ADER approach is the solution of the derivative Riemann problem, or DRP, [3], [4], a generalization of the classical Riemann problem first used by Godunov (1959) to construct a first-order upwind numerical method for hyperbolic systems. The approximate solution of the DRP requires a scheme for the leading term. Here, various possibilities are discussed, including the MUSTA approach [5],[6],[7]. I conclude this lecture with an overview of possible extensions of the ADER approach and potential applications in science and engineering. [1] Toro E F, Millington R C and Nejad L A M (2001). Towards Very High-Order Godunov Schemes. In Godunov Methods: Theory and Applications, Edited Review, Toro E F (Editor), Kluwer Academic / Plenum Publishers, pp 905-938, 2001. [2] Titarev V A and Toro E F. ADER Schemes for three-dimensional non-linear hyperbolic systems. J. Com. Phys, Vol. 204, pp 715-736, 2005. [3] Toro E. F. and Titarev V. A. Solution of the Generalised Riemann Problem for AdvectionReaction Equations. Proc. Royal Society of London A Vol. 458, pp 271-281, 2002. [4] Toro E. F. and Titarev V. A. Derivative Riemann Solvers for Systems of Conservation Laws and ADER Methods. J. Comp. Phys., Vol. 212, pp 150-165, 2006. [5] Toro E. F. (2003). Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Isaac Newton Institute for Mathematical Sciences Preprint Series NI03037-NPA , University of Cambridge, UK, June 2003. In pdf format available at: http://www.newton.cam.ac.uk/preprints2003.html [6] V A Titarev and Toro E F. MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements. Int. J. Numer. Meth. Fluids, Vol. 49, pp 117-147, 2005. [7] Toro E F and Titarev V. A. MUSTA Schemes for Systems of Conservation Laws. J. Comput. Phys., 2006 (to appear). G. Vignoli Shallow water flows: sources and well-balanced schemes The shallow water equations for a horizontal channel do not have source terms due to bottom elevation; in this case the system can be written in conservative and homogeneous form. In the case of a non-horizontal channel a source term appears in the momentum equation, see e.g. [1,2]. In this case standard shock capturing numerical methods do not work satisfactorily, and they do not reproduce well the solution for the case of steady flow. In order to have an accurate numerical scheme for the shallow water system the mathematical formulation can be given in terms of the elevation of the free surface and the discharge. By using this formulation and an appropriate high order numerical scheme it is possible to construct well balanced numerical schemes, which reproduce both steady and unsteady solutions correctly. Here we present a one-dimensional explicit high order numerical scheme which is based on the ADER approach, in the form proposed in [3]. Arbitrary order of accuracy is obtained using the ENO data reconstruction procedure and the solution of a derivative Riemann problem. The resulting numerical model is capable of reproducing both steady and unsteady solutions,smooth or discontinuous. The model can be applied to solve a number of real-life problems. ReferencesM. E. Vazquez-Cendon. Improved treatment of source terms in upwind schemes for theshallow water equations in channels with irregular geometry. Journal of ComputationalPhysics, Vol. 148, 497--526, 1999. J. G. Zhou, D. M. Causon, C. G. Mingham, and D. M. Ingram. The surface gradient methodfor the treatment of source terms in the shallow-water equations. Journal of ComputationalPhysics}, Vol. 168, 1--25, 2001. E.F. Toro and V.A. Titarev. ADER schemes for scalar non-linear hyperbolic conservationlaws with source terms in three-space dimensions. Journal of Computational Physics, Vol.202, 196--215, 2005. C. E. CastroADER DG and FV schemes for shallow water flows We are concern with ADER [1] higher order numerical methods for the time-dependent two-dimensional non-linear shallow water equations in the framework of finite volumes (FV) andDiscontinuous Galerkin (DG) finite elements methods using non-structured triangular meshes.High order in space and time is obtained by (i) a high-order spatial distribution of the solutionin each element, (ii) the solution of the Derivative Riemann Problem (DRP) [1] and (iii) anaccurate computation of the numerical flux and volume integrals. Regarding the high-orderspatial distribution of the solution, in the FV method one requires cell average reconstructions [2] at each time step; in the case of DG the high-order representation of the data is built intothe scheme to the desired order [3] and no reconstruction is needed. However, in the presenceof high gradients numerical oscillations arise in the DG case, which requires theimplementation of special procedures, none of them totally satisfactory to date. For the applications we have in mind there are three aspects which are particularly demanding(i) the devise of suitable reconstructions to avoid spurious oscillations, (ii) the treatment ofdry/wet fronts and (iii) the treatment of source terms, in particular frictional terms. All threeaspects are fundamental for applications to real problems.We assess the methods by comparing numerical solutions with exact solutions and laboratoryexperiments. We intend to apply these methods to the simulation of tsunami waves and ofdam break problems. [1] Toro E. F. and Titarev V. A. Derivative Riemann Solvers for Systems of ConservationLaws and ADER Methods. J. Comp. Phys., Vol. 212, pp 150-165, 2006. [2] Dumbser M and Käser M. Arbitrary high order non-oscillatory finite volume schemes onunstructured meshes for linear hyperbolic systems. J. Comp. Phys. Submitted. [3] M. Dumbser and C.-D. Munz. Building blocks for arbitrary high order discontinuousGalerkin schemes. J. Sci. Comp., 2006 (in press). M. DumbserApplications of Arbitrary High Order Finite Volume and DiscontinuousGalerkin Schemes on Unstructured Meshes in Two and Three Space Dimensions Previous work on arbitrary high order schemes was essentially restricted to two spacedimensions. Schwartzkopff et al. [4], e.g., successfully constructed a linear finite volume(FV) scheme of arbitrary high order of accuracy in space and time for linear two-dimensionalhyperbolic systems on Cartesian grids using Toro’s and Titarev’s ADER approach [5,6,7] forhigh order accurate one-step flux computation. Käser and Iske [3] developed the first ADERfinite volume scheme for scalar nonlinear hyperbolic conservation laws on unstructuredmeshes in two space dimensions. In our presentation we will first show an extension of the approaches presented in [3] and [4]to two and three dimensional unstructured grids and any order of accuracy, see [2]. This isachieved by a new reconstruction technique which even allows for an easy and cost-efficientincorporation of WENO reconstruction on unstructured meshes in 3D, providing a high ordernon-oscillatory shock-capturing scheme. To our knowledge, this is the first three dimensionalWENO scheme ever reported in the research literature.Second, the application of the ADER approach to the discontinuous Galerkin (DG) finiteelement framework is presented [1], with results on two and three dimensional unstructuredmeshes.Our methods are designed for solving evolution equations in the time domain in complexgeometries for long times. Furthermore, high order schemes permit good resolution ofphysical phenomena even on very coarse grids. We will show numerical convergence results of ADER-DG schemes up to 10 order of accuracy in space and time on triangular grids in2D and up to 7 order in space and time on 3D tetrahedral grids. We present convergenceresults up to sixth order of accuracy in space and time on three dimensional tetrahedral gridsfor ADER-FV schemes using 3D WENO reconstruction. For applications with large computational domains where no obstacles are present in mostparts of the computational domain, such as computational aero-acoustics (CAA), the use ofpurely unstructured grids would be far too expensive. To increase computational efficiencyfor this kind of applications, a heterogeneous domain decomposition algorithm is discussed inwhich different equations, different mesh topologies and different time-steps are coupled in afully time-accurate manner, see [8]. References:[1] M. Dumbser, C.-D. Munz. Building blocks for arbitrary high order discontinuousGalerkin schemes, Journal of Scientific Computing, 2005. in press[2] M. Dumbser, M. Käser. Arbitrary high order non-oscillatory finite volume schemes onunstructured meshes for linear hyperbolic systems, submitted to J. Comp. Phys.[3] M. Käser, A. Iske. ADER Schemes on adaptive triangular meshes for scalarconservation laws, Journal of Computational Physics, 205:486-508, 2005.[4] T. Schwartzkopff, M. Dumbser, C.-D. Munz. Fast High Order ADER Schemes forLinear Hyperbolic Equations and their Numerical Dissipation and Dispersion,Journal of Computational Physics, vol. 197, pp. 532-539 (2004)[5] V. A. Titarev, E. F. Toro. ADER: Arbitrary High Order Godunov Approach, Journal ofScientific Computing, Vol. 17, pp. 609-618 (2002)[6] V.A. Titarev and E.F. Toro. ADER schemes for three-dimensional nonlinear hyperbolicsystems. Journal of Computational Physics, 204:715-736, 2005.[7] E.F. Toro and V. A. Titarev. Solution of the generalized Riemann problem foradvection-reaction equations, Proc. Roy. Soc. London, pp. 271-281, 2002.[8] J. Utzmann, T. Schwartzkopff, M. Dumbser and C.-D. Munz. Heterogeneous DomainDecomposition for CAA, AIAA Journal, in press. V. A. TitarevHigh-order methods for nonlinear elasticityThe present research is devoted to construction and comparative study of upwind methods asapplied to the one-dimensional system of nonlinear elasticity equations [1]. We first derive asimple approach for building up exact solutions to the Riemann problem in such special casesand construct a set of test problems. Then we carry out a systematic comparative study ofrecent upwind fluxes and their comparison with some well established monotone centred andupwind fluxes. The main emphasis is given on robustness and accurate resolution of delicatefeatures of the flow such as contact waves. Finally, some selected fluxes are also used in thehigh-order framework of WENO [2] and ADER [3] schemes.[1] S.K. Godunov and E.I. Romenski. Elements of Continuum Mechanics and ConservationLaws. Kluwer Academic/ Plenum Publishers, 2003. [2] G.S. Jiang and C.W. Shu. Efficient implementation of weighted ENO schemes. J. Comput.Phys. 126:202--212, 1996. [3] E. F. Toro and V. A. Titarev. Derivative Riemann Solvers for Systems of ConservationLaws and ADER Methods. J. Comp. Phys., Vol. 212, pp 150-165, 2006. M. KäserNumerical Simulation of Seismic Wave Propagation Research on the interior structure of the earth and its geophysical properties are mainly basedon results of seismology, e.g. [1] and [6]. Today, computer simulations of the propagation ofseismic waves represent an invaluable tool for the understanding of the wave phenomena,their generation and their consequences, e.g. [2] and [5]. However, the simulation of acomplete, highly accurate wave field in realistic media with complex geometry is still a greatchallenge. Therefore, it is important to study and develop highly flexible and powerfulsimulation methods like in [4], which are not only accurate but also able to incorporatedifferent geophysical material properties, e.g. anisotropy, viscoelasticity and strongheterogeneities, earthquake rupture processes and difficult internal material boundaries aswell as topography. The so-called ADER-Discontinuous Galerkin method in [3] has theunique property of a numerical scheme that achieves arbitrarily high approximation order forthe solution of the governing system of the seismic wave equations. By the development ofsuch highly accurate algorithms and their combination with latest high performance computertechnology we aim for the solution of actual problems in numerical seismology in order toimprove ground motion prediction after strong earthquake events, to estimate local seismichazard more precisely. [1] Aki, K. and P.G. Richards: Quantitative Seismology, University Science Books, 2002. [2] Bedford, A. and D.S. Drumheller: Elastic Wave Propagation, Wiley, 1994. [3] Käser, M. and M. Dumbser: An Arbitrary High Order Discontinuous Galerkin Method forElastic Waves on Unstructured Meshes I: The Two-Dimensional Isotropic Case with ExternalSource Terms, to appear in Geophysical Journal International. [4] Käser, M. and A. Iske: ADER Schemes on Adaptive Triangular Meshes for ScalarConservation Laws, Journal of Computational Physics ,Vol.205, p.486-508, 2005. [5] Pujol, J.: Elastic Wave Propagation and Generation in Seismology, Cambrigde UniversityPress, 2003. [6] Stein, S. and M. Wysession: An Introduction to Seismology, Earthquakes and EarthStructure, Blackwell Publishing, 2003.