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In this work, we introduce a self-adaptive implicit-explicit (IMEX) time integration scheme, named IMEX-RB, for the numerical integration of systems of ordinary differential equations (ODEs), arising from spatial discretizations of partial…

Numerical Analysis · Mathematics 2025-07-28 Micol Bassanini , Simone Deparis , Francesco Sala , Riccardo Tenderini

A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where…

Numerical Analysis · Mathematics 2025-04-15 Lingyun Ding

Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. Combined with explicit…

Computational Engineering, Finance, and Science · Computer Science 2026-01-13 Christian Faßbender , Tim Bürchner , Philipp Kopp , Ernst Rank , Stefan Kollmannsberger

The incompressible Euler equations are an important model system in computational fluid dynamics. Fast high-order methods for the solution of this time-dependent system of partial differential equations are of particular interest: due to…

Numerical Analysis · Mathematics 2024-10-15 Eike Hermann Müller

We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations discretized on staggered (MAC) grids. The scheme is based on finite difference approximations…

Numerical Analysis · Mathematics 2025-12-24 Andreu Martorell , Pep Mulet , Dionisio F. Yáñez

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed…

Numerical Analysis · Mathematics 2023-01-04 Rujeko Chinomona , Daniel R. Reynolds

We propose two new classes of time integrators for stiff DEs: the implicit-explicit exponential (IMEXP) and the hybrid exponential methods. In contrast to the existing exponential schemes, the new methods offer significant computational…

Numerical Analysis · Mathematics 2016-05-11 Vu Thai Luan , Mayya Tokman , Greg Rainwater

In many applications, the governing PDE to be solved numerically contains a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is often preferred. On the other…

Numerical Analysis · Mathematics 2021-04-27 Kevin Chow , Steven J. Ruuth

In this article we design a finite volume semi-implicit IMEX scheme for the incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a time discretization technique that separates explicit and implicit terms which…

Numerical Analysis · Mathematics 2023-08-08 Michele Giuliano Carlino , Walter Boscheri

Euler--Euler or volume-averaged Navier--Stokes equations are used in various applications to model systems with two or more interpenetrating phases. Each fluid obeys its own momentum and mass equations, and the phases are typically coupled…

Numerical Analysis · Mathematics 2025-08-04 Douglas Pacheco , Richard Schussnig

We consider the development of high order asymptotic-preserving linear multistep methods for kinetic equations and related problems. The methods are first developed for BGK-like kinetic models and then extended to the case of the full…

Numerical Analysis · Mathematics 2016-03-06 Giacomo Dimarco , Lorenzo Pareschi

In this paper, we propose a new high order semi-implicit scheme for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions…

Numerical Analysis · Mathematics 2021-10-28 Sebastiano Boscarino , Jing-Mei Qiu , Giovanni Russo , Tao Xiong

We present a quantitative comparison between two different Implicit-Explicit Runge-Kutta (IMEX-RK) approaches for the Euler equations of gas dynamics, specifically tailored for the low Mach limit. In this regime, a classical IMEX-RK…

Numerical Analysis · Mathematics 2025-10-23 Giuseppe Orlando , Sebastiano Boscarino , Giovanni Russo

Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing…

Numerical Analysis · Mathematics 2020-11-24 Ross Glandon , Mahesh Narayanamurthi , Adrian Sandu

We explore a class of splitting schemes employing implicit-explicit (IMEX) time-stepping to achieve accurate and energy-stable solutions for thin-film equations and Cahn-Hilliard models with variable mobility. This splitting method…

Numerical Analysis · Mathematics 2024-05-31 Saulo Orizaga , Thomas Witelski

In simulations of fluid motion time accuracy has proven to be elusive. We seek highly accurate methods with strong enough stability properties to deal with the richness of scales of many flows. These methods must also be easy to implement…

Numerical Analysis · Mathematics 2020-10-14 Victor DeCaria , Sigal Gottlieb , Zachary J. Grant , William J. Layton

Although implicit-explicit (IMEX) methods for approximating solutions to semilinear parabolic equations are relatively standard, most recent works examine the case of a fully discretized model. We show that by discretizing time only, one…

Analysis of PDEs · Mathematics 2007-05-23 Michael Robinson

In this work we study different Implicit-Explicit (IMEX) schemes for incompressible flow problems with variable viscosity. Unlike most previous work on IMEX schemes, which focuses on the convective part, we here focus on treating parts of…

Numerical Analysis · Mathematics 2024-08-21 Gabriel R. Barrenechea , Ernesto Castillo , Douglas R. Q. Pacheco

The aim of this work is to apply a semi-implicit (SI) strategy within a Rosenbrock-type and IMEX linear multistep (LM) framework to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives.…

Numerical Analysis · Mathematics 2026-02-20 Boscarino Sebastiano , Giuseppe Izzo

Implicit-Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity of interest for…

Numerical Analysis · Mathematics 2016-10-19 Jehanzeb H. Chaudhry , J. B. Collins , John N. Shadid