The lid-driven cavity test case is widely used to validate incompressible Navier–Stokes flow solvers. However, the rigorous treatment of discontinuous corner boundary conditions remains a challenge for high-order methods. This is the main goal of the paper. We write the incompressible Navier–Stokes equations in skew-symmetric form and we impose the boundary conditions weakly which leads to boundedness without any special treatment in the corners. The continuous procedure is mimicked in the discrete setting using high-order finite difference methods in Summation-By-Parts (SBP) form complemented with weak boundary conditions using the Simultaneous Approximation Term (SAT) technique. Stability is then formally proven using the SBP-SAT framework. Numerical tests commence with a method-of-manufactured solution and the scheme is shown to be high order accurate. The lid-driven cavity test case is finally studied using a 4th order accurate scheme and the results are compared to benchmark solutions. Accurate solutions are achieved that are devoid of spurious oscillations near the top corners and the velocities remain bounded, demonstrating the unique versatility of the weak SAT boundary treatment.

We consider the Frank-Kamenetskii partial differential equation as a model for combustion in Cartesian, cylindrical and spherical geometries. Due to the presence of a singularity in the equation stemming from the Laplacian operator, we consider a specific conservative continuous formulation thereof, which allows for a discrete energy estimate. Furthermore, we consider multiple methodologies across multiple domains. On the left domain, close to the singularity, we employ the Galerkin method which allows us to integrate over time appropriately, and on the right domain we implement the finite difference method. We also derive a condition at the singularity that removes a potentially artificial boundary layer. The summation-by-parts (SBP) methodology assists us in coupling these two numerical schemes at the interface, so that we end up with a provably stable and conservative hybrid numerical scheme. We provide numerical support for the theoretical derivations and apply the procedure to a realistic case.

We show that a reformulation of the governing equations for incompressible multi-phase flow in the volume of fluid setting leads to a well defined energy rate. New nonlinear inflow-outflow and solid wall boundary conditions bound the energy rate and lead to an energy estimate in terms of only external data. The new formulation combines perfectly with summation-by-parts operators and leads to provable energy stability.

We present a comparative analysis of the continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element (FE) methods, focusing on their efficiency in solving advection–diffusion problems in the Summation By Parts–Simultaneous Approximation Term (SBP–SAT) framework. Our findings provide insights into the practical advantages and limitations of each approach, guiding future applications in high–order numerical simulations of real world applications.

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.

We present a novel solution procedure for initial boundary value problems. The procedure is based on an action principle, in which coordinate maps are included as dynamical degrees of freedom. This reparametrization invariant action is formulated in an abstract parameter space and an energy density scale associated with the space-time coordinates separates the dynamics of the coordinate maps and of the propagating fields. Treating coordinates as dependent, i.e. dynamical quantities, offers the opportunity to discretize the action while retaining all space-time symmetries and also provides the basis for automatic adaptive mesh refinement (AMR). The presence of unbroken space-time symmetries after discretization also ensures that the associated continuum Noether charges remain exactly conserved. The presence of coordinate maps in addition provides new freedom in the choice of boundary conditions. An explicit numerical example for wave propagation in 1+1 dimensions is provided, using recently developed regularized summation-by-parts finite difference operators.

We present a straightforward energy stable weak implementation procedure of open boundary conditions for nonlinear initial boundary value problems. It is mathematically identical, reduces the number of parameters and simplifies previous work and its practical implementation.

We show that a specific skew-symmetric formulation of the nonlinear terms in the compressible Navier-Stokes equations leads to an energy rate in terms of surface integrals only. No dissipative volume integrals contribute to the energy rate. We also discuss boundary conditions that bounds the surface integrals.

A stencil-adaptive SBP-SAT finite difference scheme is shown to display superconvergent behavior. As proof of concept, applied to the linear advection equation, it has a convergence rate O(Δx⁴) in contrast to a conventional scheme, which converges at a rate O(Δx³).

Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics along a world-line parameter, instead of physical time directly, we retain manifest translation symmetry and conservation of the associated continuum Noether charge. A non-equidistant time discretization emerges dynamically, realizing a form of automatic adaptive mesh refinement (AMR), guided by the system symmetries. Using appropriately regularized summation by parts finite difference operators, the continuum Noether charge, defined via the Killing vector associated with translation symmetry, is shown to be exactly preserved in the interior of the simulated time interval. The convergence properties of the approach are demonstrated with two explicit examples.