Kenneth Duru

Kenneth Duru, Alice-Agnes Gabriel, Gunilla Kreiss

We develop a provably energy stable discontinuous Galerkin spectral element method (DGSEM) approximation of the perfectly matched layer (PML) for the three and two space dimensional (3D and 2D) linear acoustic wave equations, in first order form, subject to well-posed linear boundary conditions. First, using the well-known complex coordinate stretching, we derive an efficient un-split modal PML for the 3D acoustic wave equation. Second, we prove asymptotic stability of the continuous PML by deriving energy estimates in the Laplace space, for the 3D PML in a heterogeneous acoustic medium, assuming piece-wise constant PML damping. Third, we develop a DGSEM for the wave equation using physically motivated numerical flux, with penalty weights, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Fourth, to ensure numerical stability when PML damping is present, it is necessary to systematically extend the numerical numerical fluxes, and the inter-element and boundary procedures, to the PML auxiliary differential equations. This is critical for deriving discrete energy estimates analogous to the continuous energy estimates. Finally, we propose a procedure to compute PML damping coefficients such that the PML error converges to zero, at the optimal convergence rate of the underlying numerical method. Numerical experiments are presented in 2D and 3D corroborating the theoretical results.

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.

Dougal Stewart, Kenneth Duru

A recent study by Gassner et al. [J. Sci. Comput. 90:79 (2022)] demonstrates that local energy stability--that is, ensuring the asymptotic numerical growth rate does not exceed the continuous growth rate--is crucial for achieving accurate numerical simulations of nonlinear conservation laws. While nonlinear entropy stability is necessary for numerical stability (i.e., ensuring the boundedness of nonlinear numerical solutions), local energy stability is essential to prevent unresolved high-frequency wave modes from dominating the simulation. Currently, it remains an open question whether high-order numerical methods for nonlinear conservation laws can be simultaneously entropy-stable and locally energy-stable. In this work, we examine the local energy-stability properties of recently developed entropy-stable, high-order accurate dual-pairing (DP) SBP methods, as introduced by Duru et al. [arXiv: 2411.06629], for nonlinear conservation laws. Our analysis indicates that the entropy-stable volume upwind filter inherent in these methods can ensure local energy stability. This approach offers a novel numerical strategy for designing reliable high-order methods for nonlinear conservation laws that are provably entropy-stable and locally energy-stable. The theoretical findings are supported by numerical experiments involving the inviscid Burgers equation and nonlinear shallow water equations, in 1D and 2D. Furthermore, we present accurate numerical simulations of 2D barotropic shear instability, with fully developed turbulence, demonstrating the efficiency of the DP SBP method in resolving turbulent scales.

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.