Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.03889

A New Class of $A$ Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

Hendrik Ranocha·Jan Nordström
View original source ↗Read source document ↗Export metadata ↓

Abstract

Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of $A$ stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hendrik Ranocha, Jan Nordström. 2021-02-11. A New Class of $A$ Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions. https://doi.org/10.1007/s10915-021-01454-7

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Stability of Block Eliminations and Additive Modifications

The block elimination with additive modifications (BEAM) method was recently proposed as a alternative to LU with partial pivoting requiring less communication. Because of the novelty of BEAM, the existing theoretical analysis is lacking. To that end, we analyze both the numerical stability of the underlying block LU factorization and the effects of additive modifications. For the block LU factorization, we are able to improve the previous results of Demmel et al. from being cubic in the element growth to merely quadratic. Furthermore, we propose an alternative measure of element growth that is better aligned with block LU; this new measure of growth allows our analysis to apply to matrices that cannot be factored with pointwise LU. In the second part, we analyzed the modifications produced by BEAM and the effect they have on the condition number and growth factor. Finally, we show that BEAM will not apply any modifications in some cases that regular block LU can safely factor.

math.NA↗
arXiv2026

Efficient Rigorous Continuation via Chebyshev Series Expansion I

We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted $\ell^1$ space, giving a finer control than the typical $C^0$-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn--Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada--Kawasaki--Teramoto system.

math.NA↗
arXiv2026

Efficient iterative techniques for solving tensor problems with the T-product

This paper develops two efficient iterative methods for solving tensor equations under the T-product framework. For T-symmetric positive definite tensor equations of the form $\mathcal{C} \star \mathcal{X} = \mathcal{D}$, we propose a conjugate-gradient-type algorithm that generates orthogonal residual and $\mathcal{C}$-orthogonal direction sequences, ensuring convergence within a finite number of steps. For general consistent tensor equations, we extend the method using a normal-equation transformation, and further adapt it to handle inconsistent systems by solving a least-squares minimization problem. Key advantages include direct tensor-based computations without explicit matrix expansion, rigorous finite-step convergence proofs, and the ability to obtain minimal Frobenius norm solutions. Numerical experiments on synthetic data, benchmark images, and video sequences demonstrate that the proposed algorithms achieve high precision with low computational time, confirming their practicality for large-scale multidimensional problems.

math.NA↗