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Jan Nordström

Publications and source records attributed to Jan Nordström.

At least 19 recordsLinked to original sources
arXiv2026

Towards provable energy-stable overset grid methods using sub-cell summation-by-parts operators

Overset grid methods handle complex geometries by overlapping simpler, geometry-fitted grids to cover the original, more complex domain. However, ensuring their stability---particularly at high orders---remains a theoretical challenge: although overset grid methods perform robustly in extensive practical use, general stability proofs are not available. In this work, we address this gap by developing a discrete counterpart to the recent well-posedness analysis of Kopriva, Gassner, and Nordström for continuous overset domain initial-boundary-value problems. To this end, we introduce the novel concept of sub-cell summation-by-parts (SBP) operators. These discrete derivative operators mimic integration by parts at a sub-cell level. By exploiting this sub-cell SBP property, we develop provably conservative and energy-stable overset grid methods for fixed one-dimensional overset domains that do not change with time or under grid refinement, providing a step toward stability proofs for overset grid methods based on the energy method.

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arXiv2026

Global Bounds for the Error in Solutions of Linear Hyperbolic Systems due to Inaccurate Boundary Geometry

Meshes approximate the boundaries of a geometry when the boundaries are curved. The accuracy of the mesh then affects the error of computations of initial boundary value problems for partial differential equations, especially when using high order methods. Here, we derive global estimates for the error in solutions of linear hyperbolic systems due to inaccurate boundary geometry. We show that the error is bounded by data and bounded in time when the solutions in the true and approximate domains are bounded. Just evaluating boundary data at the correct location has a secondary effect on the error, whereas the primary errors are from the Jacobian and metric terms. In two space dimensions, specifically, we show that to lowest order the errors are proportional to the errors in the boundary curve locations and their derivatives. Therefore, high order accuracy computations cannot be obtained unless the mesh is also high order. The results illustrate the importance of accurately approximating boundaries and should be helpful guides for high-order mesh generation for advection-dominated problems and the design of optimization algorithms for boundary approximations.

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arXiv2026

Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators

The Active Flux method is a numerical method for conservation laws using a combination of cell averages and point values as independent degrees of freedom, based on ideas from finite volumes and finite differences. This unusual mix has been shown to work well in many situations. We expand the theoretical justifications of the Active Flux method by analyzing it from the point of view of summation-by-parts (SBP) operators, which are routinely used to analyze finite difference, finite volume, and finite element schemes. We investigate in what type of setting the Active Flux method can be formulated using classical or degenerate SBP operators, yielding a first and novel approach for showing the energy stability of the Active Flux method. We present the analysis for the one-dimensional scalar linear advection equation with periodic boundary conditions on a uniform grid.

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arXiv2026

A well posed and stable canonical evaporation model problem for phase-change in two-phase flows

We formulate a well posed interface formulation for canonical one-dimensional evaporation two-phase model problems (the Stefan and Sucking problems) commonly used to validate production codes. We focus on the interface between the vapor and the liquid and derive conditions leading to an energy bound and well-posedness. Next, by mimicking the continuous analysis, we discretize using high order accurate numerical methods on summation-by-parts form, impose the interface conditions weakly and prove energy stability.

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arXiv2026

Summation-by-parts operators for general function spaces: optimal nodes

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 dimension. We show that the same principle extends to general (possibly non-polynomial) function spaces: an associated generalised Gauss-Lobatto quadrature provides the optimal nodes and weights for the SBP formulation. We present an algorithm for computing these quadrature rules, demonstrate their accuracy and efficiency across a range of function spaces, and illustrate their use in solving initial boundary value problems.

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arXiv2026

A high order accurate and energy stable continuous Galerkin framework on summation-by-parts form for the incompressible Navier-Stokes equations

This paper presents a high-order accurate Continuous Galerkin Finite Element Method (CGFEM) for solving the initial boundary value problems governed by the Incompressible Navier-Stokes (INS) equations. We discretize the INS equations using the CGFEM approach in Summation-By-Parts (SBP) form. Lagrange polynomials of up to 4th order are employed. The boundary conditions are imposed weakly using the Simultaneous Approximation Term (SAT) technique, which accommodates discontinuous boundary data without special treatment. The resulting SBP-SAT formulation guarantees an energy stable discretization. The efficiency of the proposed framework is demonstrated by solving a series of numerical tests. Initially, the Method of Manufactured Solutions (MMS) is employed to demonstrate 4th order convergence. Subsequently, the 4th order accurate scheme is applied to a classical benchmark problem featuring discontinuous boundary conditions: the lid-driven cavity flow over a wide range of Reynolds numbers. Accurate and oscillation-free solutions are achieved even in the vicinity of the discontinuous top corner boundaries. Lastly, a canonical backward-facing step flow problem is solved, where accuracy and efficiency are demonstrated.

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arXiv2026

From Exact Space-Time Symmetry Conservation to Automatic Mesh Refinement in Discrete Initial Boundary Value Problems

In this contribution we present recent developments in the formulation and solution of Initial Boundary Value Problems (IBVPs). Building upon a modern variational action formulation of classical dynamics, we treat Initial Boundary Value Problems directly on the action level, bypassing governing equations. We show that by including coordinate maps as dynamical degrees of freedom together with propagating fields two key results emerge. Space-time symmetries remain protected even after discretization, leading to an exact conservation of Noether charges even for discrete IBVPs. The dynamical nature of the coordinate maps leads to an adjustment of space-time resolution, guided by Noether charge conservation, realizing a form of automatic adaptive mesh refinement. We stress that as long as SBP operators are used for the discretization, our results are independent of whether the dynamics are solved on the action or governing equation level and hold in particular also at high order. As proof-of-principle for our approach we present its application to scalar wave-propagation in 1+1 dimensions.

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arXiv2026

Numerical boundary flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries

We present a strategy for interpreting nonlinear, characteristic-type penalty terms as numerical boundary flux functions that provide provable bounds for solutions to nonlinear hyperbolic initial boundary value problems with open boundaries. This approach is enabled by recent work that found how to express the entropy flux as a quadratic form defined by a symmetric boundary matrix. The matrix formulation provides additional information for how to systematically design characteristic-based penalty terms for the weak enforcement of boundary conditions. A special decomposition of the boundary matrix is required to define an appropriate set of characteristic-type variables. The new boundary fluxes are directly compatible with high-order accurate split form discontinuous Galerkin spectral element and similar methods and guarantee that the solution is entropy stable and bounded solely by external data. We derive inflow-outflow boundary fluxes specifically for the Burgers equation and the two-dimensional shallow water equations, which are also energy stable. Numerical experiments demonstrate that the new nonlinear fluxes do not fail in situations where standard boundary treatments based on linear analysis do.

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arXiv2026

A high order accurate and provably stable fully discrete continuous Galerkin framework on summation-by-parts form for advection-diffusion equations

We present a high-order accurate fully discrete numerical scheme for solving Initial Boundary Value Problems (IBVPs) within the Continuous Galerkin (CG)-based Finite Element framework. Both the spatial and time approximation in Summation-By-Parts (SBP) form are considered here. The initial and boundary conditions are imposed weakly using the Simultaneous Approximation Term (SAT) technique. The resulting SBP-SAT formulation yields an energy estimate in terms of the initial and external boundary data, leading to an energy-stable discretization in both space and time. The proposed method is evaluated numerically using the Method of Manufactured Solutions (MMS). The scheme achieves super-convergence in both spatial and temporal direction with accuracy $\mathcal{O}(p+2)$ for $p\geq 2$, where $p$ refers to the degree of the Lagrange basis. In an application case, we show that the fully discrete formulation efficiently captures space-time variations even on coarse meshes, demonstrating the method's computational effectiveness.

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arXiv2025

Linear and Nonlinear Boundary Conditions: What's the difference?

In previous work, we derived new energy and entropy stable open boundary conditions and implementation procedures for linear and nonlinear initial boundary value problems. These boundary procedures results in estimates bounded by external data only. Interestingly, these new boundary conditions generalize the well known classical characteristic boundary conditions for linear problems to the nonlinear setting. We discuss the similarities and differences between these two boundary procedures and point out the advantages with the new procedures. In particular we show that the new boundary conditions bound both linear and nonlinear initial boundary value problems and can be implemented both strongly and weakly.

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arXiv2024

An Energy Stable Nonlinear Incompressible Multi-Phase Flow Formulation

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. Weak nonlinear inflow-outflow and solid wall boundary conditions complement the development and lead to an energy estimate in terms of external data. The new formulation combined with summation-by-parts operators lead to provably nonlinear energy stability.

math.AP↗
arXiv2024

Energy Bounds for Discontinuous Galerkin Spectral Element Approximations of Well-Posed Overset Grid Problems for Hyperbolic Systems

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.

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arXiv2024

An optimization-based construction procedure for function space based summation-by-parts operators on arbitrary grids

We introduce a novel construction procedure for one-dimensional summation-by-parts (SBP) operators. Existing construction procedures for FSBP operators of the form $D = P^{-1} Q$ proceed as follows: Given a boundary operator $B$, the norm matrix $P$ is first determined and then in a second step the complementary matrix $Q$ is calculated to finally get the FSBP operator $D$. In contrast, the approach proposed here determines the norm and complementary matrices, $P$ and $Q$, simultaneously by solving an optimization problem. The proposed construction procedure applies to classical SBP operators based on polynomial approximation and the broader class of function space SBP (FSBP) operators. According to our experiments, the presented approach yields a numerically stable construction procedure and FSBP operators with higher accuracy for diagonal norm difference operators at the boundaries than the traditional approach. Through numerical simulations, we highlight the advantages of our proposed technique.

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arXiv2024

Exact symmetry conservation and automatic mesh refinement in discrete initial boundary value problems

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.

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arXiv2024

Summation-by-parts operators for general function spaces: The second derivative

Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.

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arXiv2023

The crucial role of Lagrange multipliers in a space-time symmetry preserving discretization scheme for IVPs

In a recently developed variational discretization scheme for second order initial value problems ( J. Comput. Phys. 498, 112652 (2024) ), it was shown that the Noether charge associated with time translation symmetry is exactly preserved in the interior of the simulated domain. The obtained solution also fulfils the naively discretized equations of motions inside the domain, except for the last two grid points. Here we provide an explanation for the deviations at the boundary as stemming from the Lagrange multipliers used to implement initial and connection conditions. We show explicitly that the Noether charge including the boundary corrections is exactly preserved at its continuum value over the whole simulation domain, including the boundary points.

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