Kenneth Duru

MSI Fellow

Dr Duru is a computational and applied mathematician. He is currently a MSI Fellow at the Mathematical Sciences Institute, Australian National University, where he bears the torch of numerical analysis of partial differential equations (PDEs). He studied mathematics and statistics at the University of Calabar, Nigeria. In 2012, he earned his PhD in Scientific Computing (major Numerical Analysis) from Uppsala University, Sweden, under the supervision of Prof. Gunilla Kreiss. Dr Duru’s research lie at the interfaces of mathematical analysis, numerical analysis and high-performance computing (HPC), and contributes to the mathematical foundation of numerical methods and simulation tools for the solution of PDEs modelling complex real world problems such as earthquakes. His research mission is dedicated to the development of provably accurate, robust, flexible, hybrid and adaptive framework, based on a rigorous mathematical foundation, for efficient numerical simulation of multi-scale and multi-physics wave phenomena on modern and next generation supercomputers.

He was a member of the EU funded consortium, ExaHyPE, dedicated to the development of an exa-scale hyperbolic PDE simulation engine for exa-scale supercomputers. He spent more than three years at the Stanford University, as a postdoc, working with Prof. Eric Dunham. At Stanford University, his research focused on developing the simulation code WaveQLab3D, and high order accurate and provably stable finite difference methods for waves and earthquake rupture dynamics in 3D heterogeneous and geometrically complex Earth models.

Dr Duru recently joined the Southern Ocean Project where he collaborates with Prof Hrvoje Tkalčić at the ANU's Research School of Earth Sciences, and contributes to seismic wavefield and source modeling,  including submarine landslide and tsunami modelling.

Research interests

Dr Duru's research interest encompasses all components of the development of numerical algorithms and simulation tools for partial differential equations (PDEs):

  • Convergent  volume discretisation in complex geometries using high order methods (finite difference methods, discontinuous Galerkin methods, finite element and spectral methods).

  • Mathematical and numerical analysis of (hyperbolic) PDEs.

  • Development and analysis of perfectly matched layers (PML)  – to absorb outgoing waves at domain boundaries.

  • Efficient implementation of scalable simulation codes on modern, heterogeneous many-core HPC platforms.

  • Numerical simulations of real-world wave propagation problems.

Some ongoing and future projects are listed below for prospective research/PhD students and postdocs.

High fidelity and extreme-scale methods for multi-scale wave propagation problems

Multi-physics/multi-scale problems, simultaneously coupling different physical phenomena at different scales in space and time, occur in many applications. One example is simulations of coupled elastic deformation and magma flows during volcanic eruptions, and the associated waves, as well as earthquake source process, coupling frictional failure on the fault to wave propagation off the fault, in the elastic medium. To effectively simulate such problems, advanced methods and hybrid solvers coupling different numerical methods with dynamic adaptivity are essential, as well as the availability of sufficient computational resources.

The main aims of this topic include:

  • Develop provably stable hybrid solvers that couple DG methods to SBP FD methods for efficient numerical simulation of multi-scale and/or multi-physics wave propagation problems.

  • Provide a rigorous analysis of a stable and high order accurate coupling with nontrivial boundary conditions for a class of DG, finite volume and FD approximations with efficient explicit or semi-explicit time discretisation.

  • Implement the numerical methods in a scalable framework such as ExaHyPE or WaveQLab.


Perfectly matched layer (PML), absorbing layers and absorbing boundary conditions

The main features of propagating waves is that they can propagate long distances relative to their characteristic length-scale, the wavelength. For numerical simulations, it is precisely this essential feature of waves, the radiation of waves to far field, that leads to the greatest difficulties. The perfectly matched layer (PML) is a method which, when stable, can provide a domain truncation which is convergent with increasing layer width/damping. The difficulties in using the PML are primarily associated with stability, which can be present at the continuous level or be triggered by numerical approximations.  The mathematical and numerical analysis of the PML for hyperbolic wave propagation problems has been area of active research. It is now possible to construct stable and high order accurate numerical wave solvers using  summation-by-parts finite difference methods, continuous or discontinuous Galerkin finite element methods.

The main aim of this topic is to answer some of the following important questions:

  • How can PML be adapted to the presence of surface and interface waves?

  • Can we derive a stable PML for general heterogeneous isotropic or anisotropic elastic media?

  • Can we derive a stable and effective absorbing layer for nonlinear problems, with examples from nonlinear aero-acoustics and general relativity?

Dynamic earthquake rupture simulations 

Earthquakes are catastrophic natural events that can take a heavy toll on our societies, through life and property losses. By improving our understanding on how earthquakes nucleate and propagate, we may make progress towards mitigating their negative impact. Insights from modelling and simulations may help in the development of better seismic hazard models as well as in the construction of more efficient earthquake early warning systems. However, earthquakes present us with some of the most difficult mathematical, numerical and computational challenges. One main objective under this topic is to help seismologists correctly model earthquake source processes to better quantify and constrain seismic hazard at frequencies relevant to engineering applications.

This topic will address some of these important mathematical and numerical challenges:

  • Prove well/ill-posedness of the IBVP for a class of modern friction laws for crustal rocks.

  • Derive stable numerical methods to efficiently solve quasi-static elasticity coupled to nonlinear friction laws and enable efficient simulation of earthquake sequences.

  • Derive stable numerical methods to efficiently couple quasi-static elasticity to dynamic elasticity subject to nonlinear friction laws.

Space-time adaptive DG for Einstein’s equations in second order form.

Second order hyperbolic systems often describe problems where wave propagation is dominant. Typical examples are the acoustic wave equation, elastic wave equation, and the Einstein’s equation of general relativity. However, many solvers for wave equations and the Einstein’s equations are designed for first order systems. In particular multi-domain spectral methods which are increasingly becoming attractive because they are optimal, in terms of efficiency and accuracy, are commonly implemented as first-order systems. That is, the system of second order hyperbolic equations

are first reduced to a system of first order hyperbolic PDEs before numerical approximations are introduced. The main reason being that the theory and numerical methods to solve hyperbolic PDEs are well developed for first order hyperbolic systems, and less developed for second order systems.

There are many advantages to solving the equations in second order form. The reduction of the equation to first-order hyperbolic form, has the disadvantage of introducing auxiliary variables with their constraints and boundary conditions. For example, in the harmonic description of general relativity, the Einstein’s equations are a system of 10 curved space second order wave equations, while the corresponding reduction to first order systems will involve about 60 equations. The reduction to first-order form is also less attractive from a computational point of view considering the efficiency and accuracy of numerical approximation.

It has been a long held ambition of the computational relativity community to develop the theory and and provably stable numerical techniques for second order hyperbolic systems such that Einstein’s equations can be solved efficiently. This has proven to be an incredibly difficult task. In particular, it appears to be more difficult to guarantee stability for naturally second order systems than for first order reductions of them.

The overall goal in this project is to design a new class of spectrally accurate and robust numerical algorithms to solve Einstein’s equations in second order form, and simulate relativistic astrophysical wave phenomena at an unprecedented precision never achieved before.


  • K. Duru,  S. Wang and K. Wiratama, A conservative and energy stable discontinuous spectral element method for the shifted wave equation in second order form (2021), arXiv:2107.05770, submitted

  • K. Duru, F. Frederick and C. WIlliams, Upwind summation by parts finite difference methods for large scale elastic wave simulations in 3D complex geometries (2021), arXiv:2011.02600, submitted.

  • K. Duru, L. Rannabauer, O. K. A. Ling, A.-A. Gabriel, H. Igel, and M. Bader, A stable discontinuous Galerkin method for linear elastodynamics in 3D geometrically complex elastic solids using physics based numerical fluxes (2020), arXiv:1907.02658, submitted.

  • K. Wiratama, S. Roberts and K. Duru, Weak imposition of boundary conditions for the gauge formulation of the incompressible Navier-Stokes equations (2020), The ANZIAM Journal of Mathematics,  accepted May 2021

  •  K Duru, L. Rannabauer, A.-A. Gabriel and H. Igel,  A new discontinuous Galerkin  method  for elastic waves with physically motivated numerical fluxes (2020), arXiv:1802.06380, Journal of Scientific Computing (2021) 88:51

  • L. Rannabauer,  A.-A. Gabriel, G. Kreiss and M. Bader,  A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form (2020) Numerische Mathematik,

  • A. Reinarz, D. E. Charrier, M. Bader, L. Bovard, M. Dumbser, K. Duru, F. Fambri, A.-A. Gabriel, J.-M. Gallard, S. K\"oppel, L. Krenz, L. Rannabauer, L. Rezzolla, P. Samfass, M. Tavelli, T. Weinzierl,  ExaHyPE: An Engine for Parallel Dynamically Adaptive Simulations of Wave Problems (2020), Computer Physics Communications,

  • K. Duru,  K. L Allison, M. Rivet, and E. M. Dunham, Dynamic Rupture and Earthquake Sequence Simulations Using the Wave Equation in Second-Order Form,  Geophys. J. Int.  219 (2019), 796--815.

  • K. Duru, A.-A. Gabriel and G. Kreiss,  On  energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for  the wave equation, Comput.  Methods  Appl.  Mech.  Engrg.  350  (2019)  898--937.

  • Lion Krischer, Yongki Andita Aiman,  Timothy Bartholomaus, Stefanie Donner,  Martin van Driel, Kenneth Duru, Kristina Garina, Kilian Gessele, Tomy Gunawan,  Sarah Hable,  C\'eline Hadziioannou, Mathijs Koymans,  John Leeman,  Fabian Lindner,  Angel Ling,  Tobias Megies,  Ceri Nunn,  Ashim Rijal,  Johannes Salvermoser,  Sujania Talavera Soza, Carl Tape,  Taufiq Taufiqurrahman, David Vargas,  Joachim Wassermann,  Florian W\"olfl,  Mitch Williams,  Stephanie Wollherr,  Heiner Igel, Seismo-live: An Educational Online Library of Jupyter Notebooks for Seismology, Seismological Research Letters, V 89/6, 2413-2419, doi:10.1785/0220180167  (2018).  

  • R.A. Harris, M. Barall, B. Aagaard, S. Ma, D. Roten, K. Olsen, B. Duan, D. Liu, B. Luo, K. Bai, J.P. Ampuero, Y. Kaneko, A. Gabriel, K. Duru, T. Ulrich, S. Wollherr, Z. Shi, E. Dunham, S. Bydlon, Z. Zhang, X. Chen, S.N. Somala, C. Pelties, J. Tago, V. Cruz-Atienza, J. Kozdon, E. Daub, K. Aslam, Y. Kase, A suite of exercises for verifying 3D dynamic earthquake rupture codes,   Seismological Research Letters Seismological Research Letters  Volume 89, Number 3.

  • K. Duru and E. M. Dunham, Dynamic earthquake rupture simulations on nonplanar faults embedded in 3D geometrically complex, heterogeneous Earth models, Journal of Computational Physics 305 (2016) 185–207. pdf 

  • K. Duru, The role of numerical boundary procedures in the stability of perfectly matched layers,  SIAM Journal on Scientific Computing,  Vol. 38, No. 2, pp. A1171-A1194 (2016). pdf

  • K. Duru, J. E. Kozdon and G. Kreiss, Boundary conditions and  stability of a perfectly matched layer for the elastic wave equation in first order form,  Journal of Computational Physics 303 (2015) 372–395. pdf

  • K. Duru and G. Kreiss, Boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form, SIAM Journal on Numerical Analysis Vol. 52, No. 6, pp. 2883-2904,  (2014). pdf

  • K. Duru and K. Virta, Stable and high order accurate difference methods for the elastic wave equation in discontinuous media, Journal of Computational Physics, DOI: 10.1016/ (2014). pdf

  • K. Duru, G. Kreiss and K. Mattsson, Accurate and stable boundary treatment for the elastic wave equations in second order formulation, SIAM Journal on Scientific Computing,  36(6), A2787-A2818 (2014). pdf

  • K. Duru and G. Kreiss, Numerical interaction of boundary waves with perfectly matched layers in two space dimensional elastic waveguides, Wave Motion, DOI: 10.1016/j.wavemoti.2013.11.002, (2013). pdf

  • K. Duru, A perfectly matched layer for the wave equation in heterogeneous and layered media, Journal of Computational Physics, DOI: 10.1016/, (2013). pdf

  • K. Duru and G. Kreiss, Efficient and stable perfectly matched layers for CEM, Applied Numerical Mathematics, DOI: 10.1016/j.apnum.2013.09.005, (2013). pdf

  • G. Kreiss and K. Duru, Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation, BIT Numerical Mathematics, DOI 10.1007/s10543-013-0426-4 (2013). pdf

  • K. Duru and G. Kreiss, On the accuracy and stability of the perfectly matched layer in transient waveguides, Journal of Scientific Computing, DOI 10.1007/s10915-012-9594-7 (2012). pdf

  • K. Duru and G. Kreiss, A well-posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation, Communication in Computational Physics 11 , p 1643-1672, (2012). pdf

  • K. Duru and G. Kreiss, Stable perfectly matched layers for the Schrodinger equations , Numerical mathematics and advanced applications: 2009, pp, 287-295, Springer–Verlag, Berlin, (2010). pdf

  • M. Khan, Q. Abbas, K. Duru, Magnetohydrodynamic flow of a Sisko fluid in annular pipe: A numerical study International Journal for Numerical Methods in Fluids Vol 62, issue 10, p 1169–1180, (2010). pdf

K. Duru, K. Mattsson, G. Kreiss, Stable and conservative time propagators for second order hyperbolic systems, Tech. Report, 2011-008, Dept. of Infor. Tech. Uppsala University (2011). pdf