Kenneth Duru

Dougal Stewart, Nathan Lee, Kenneth Duru

Robust and convergent high-order numerical methods for solving partial differential equations are highly attractive due to their efficiency on modern and next-generation hardware architectures. However, designing such methods for nonlinear hyperbolic conservation laws remains a significant challenge. In this work, we introduce a framework based on dual-pairing (DP) and upwind summation-by-parts (SBP) finite difference (FD) and discontinuous Galerkin (DG) finite element methods, aimed at achieving accurate and robust numerical approximations of nonlinear conservation laws. The framework ensures entropy consistency and features an intrinsic high-order accurate filter designed to detect and resolve regions where the solution is poorly captured or discontinuities are present. The DP SBP FD/DG operators form a dual pair of discrete derivative operators that collectively preserve the SBP property. Furthermore, these operators are constructed to be upwind, allowing them to incorporate dissipation within the elements themselves.This contrasts with traditional SBP and collocated DG spectral element methods, which typically induce dissipation solely through numerical fluxes at element interfaces. Our framework facilitates the systematic combination of DP SBP FD/DG operators with skew-symmetric and upwind flux splitting techniques. This integration enables the development of robust, high-order accurate schemes for nonlinear hyperbolic conservation laws.

Kenny Wiratama, Kenneth Duru, Yunho Kim

In this paper, we propose the first of its kind space-time dual-pairing summation by parts (DP-SBP) numerical framework for forward and adjoint wave propagation problems. This novel approach enables us to achieve spatial and temporal high order accuracy while naturally introducing dissipation in time. Within this framework, initial and boundary conditions are weakly imposed using the simultaneous approximation term (SAT) technique. Fully discrete energy estimates are derived, ensuring the stability of the resulting numerical scheme. Furthermore, the proposed space-time numerical framework allows us to construct adjoint consistent fully discrete numerical approximations, which can be applied to solve inverse wave propagation problems. We provide numerical experiments in one and two spatial dimensions to verify the theoretical analysis and demonstrate convergence of numerical errors.

Dean Muir, Kenneth Duru, Stuart Hudson et al.

We present a robust and accurate numerical method for the anisotropic diffusion equation in curvilinear coordinates. This study extends the recent work [Muir et al., Computer Physics Communications, 2025] for solving the anisotropic diffusion equation in magnetic fields from Cartesian meshes to to curvilinear coordinates and complex geometries. The method uses summation by parts with simultaneous approximation terms for computing the diffusion perpendicular to field lines. The diffusion along field lines is computed using a penalty approach, similar to a simultaneous approximation term, but applied across the volume. To extend the method to complex geometry we use a multi-block approach with piecewise smooth structured meshes. That is, the domain is split into sub-grids, with locally adjacent boundaries coupled weakly using penalties. We prove the semi-discrete stability for the curvilinear implementation by deriving discrete energy estimates. The approach is verified though a number of numerical tests, which demonstrate the convergence properties of the method in multi-domain approach. Finally, we present a qualitative result generated in complex geometry and magnetic field, which is generated by the Stepped Pressure Equilibrium Code.

Timothy C. Andrews, Kenneth Duru, David Lee

This paper introduces a new application of the perfectly matched layer (PML) for mitigating model top wave reflections in geophysical fluid models. Typically, a strong Laplacian or Rayleigh damping sponge layer is used near the upper boundary, but these often need many vertical levels or a high model top to be sufficiently effective. An advantage of the PML is that, at the continuous level, it is free of wave reflection at the onset of the damping layer. This enables the PML to be effective even with a thin damping layer. We derive PMLs for the linear and nonlinear versions of the Boussinesq equations, which are a simplified model for vertical dynamics in the atmosphere. In the nonlinear system, we define a novel PML that damps perturbations from a hydrostatically balanced reference state. We approximate the PML equations using the compatible finite element method for numerical experiments. First, tests with the linear Boussinesq system show that the PML is more effective than a typical sponge layer in absorbing acoustic waves near the model top. Next, tests in the nonlinear system show that i) the PML can damp acoustic waves even when they are under-resolved by the time discretisation, and ii) the PML can avoid the standing wave pattern caused by model top reflection of orographic gravity waves. We propose that the PML is worth further development and investigation as a sponge layer alternative in dynamical cores for atmospheric modelling.