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We develop a family of stabilized backward differentiation formula (sBDF) schemes of orders one through four for semilinear parabolic equations. The proposed methods are designed to achieve three properties that are rarely available…
Kernel-based approach to operator approximation for partial differential equations has been shown to be unconditionally stable for linear PDEs and numerically exhibit unconditional stability for non-linear PDEs. These methods have the same…
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard…
We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by…
High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often…
The Shifted Boundary Method (SBM) trades some part of the burden of body-fitted meshing for increased algebraic complexity. While the resulting linear systems retain the standard $\mathcal{O}(h^{-2})$ conditioning of second-order operators,…
Unfitted boundary methods are widely used to numerically solve partial differential equations (PDEs) on irregular domains, avoiding the computational burden of generating boundary-conforming grids. In the finite-difference framework,…
We analyze the stability and functional superconvergence of discretizations of diffusion problems with the narrow-stencil second-derivative generalized summation-by-parts (SBP) operators coupled with simultaneous approximation terms (SATs).…
In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid…
This paper presents a novel robust trajectory optimization method for constrained nonlinear dynamical systems subject to unknown bounded disturbances. In particular, we seek optimal control policies that remain robustly feasible with…
In this work we propose a simple but effective high order polynomial correction allowing to enhance the consistency of all kind of boundary conditions for the Euler equations (Dirichlet, characteristic far-field and slip-wall), both in 2D…
We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller…
We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-)symmetric form, thus characterizing (skew-)self-adjoint and unitary operators by means of self-ortho-gonal subspaces. By orthogonality…
Gauss-Lobatto quadrature nodes and weights are optimal for closed summation-by-parts (SBP) formulations based on polynomial approximation spaces in the sense that for a prescribed function space they yield an SBP operator of minimal…
We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial…
A computational PDE-constrained optimization approach is proposed for optimal trajectory planning under uncertainty by means of an associated Schroedinger Bridge Problem (SBP). The proposed SBP formulation is interpreted as the mean-field…
Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-$ \mathsf{E} $ operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However,…
This paper is concerned with the construction of high order schemes on irregular grids for balance laws, including a discussion of an a-posteriori error indicator based on the numerical entropy production. We also impose well-balancing on…
In a refined Sobolev scale, we investigate an elliptic boundary-value problem with additional unknown functions in boundary conditions for which the maximum of orders of boundary operators is grater than or equal to the order of the…
We study the robust output regulation of linear boundary control systems by constructing extended systems. The extended systems are established based on solving static differential equations under two new conditions. We first consider the…