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We present a family of integral equation-based solvers for the heat equation, reaction-diffusion systems, the unsteady Stokes equation and the incompressible Navier-Stokes equations in two space dimensions. Our emphasis is on the…

Numerical Analysis · Mathematics 2025-12-01 Jun Wang , Jie Su , Leslie Greengard , Shidong Jiang , Shravan Veerapaneni

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an…

Numerical Analysis · Mathematics 2020-03-27 Y. Maday , O. Mula

Recently, a new framework to compute the photoionization rate in streamer discharges accurately and efficiently using the integral form and the fast multipole method (FMM) was presented. This paper further improves the efficiency of this…

Plasma Physics · Physics 2021-12-21 Bo Lin , Chijie Zhuang

In this contribution we develop an efficient reduced order model for solving parametrized linear-quadratic optimal control problems with linear time-varying state system. The fully reduced model combines reduced basis approximations of the…

Numerical Analysis · Mathematics 2024-08-29 Hendrik Kleikamp , Lukas Renelt

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…

Numerical Analysis · Mathematics 2019-10-28 Jérôme Droniou , Robert Eymard , T. Gallouët , R. Herbin

In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based…

Optimization and Control · Mathematics 2022-08-03 Giacomo Borghi , Michael Herty , Lorenzo Pareschi

A collection of codes (in MATLAB & Fortran 77), and examples, for solving reaction-diffusion equations in one and two space dimensions is presented. In areas of the mathematical community spectral methods are used to remove the stiffness…

Numerical Analysis · Mathematics 2018-10-18 Richard V. Craster , Roberto Sassi

A method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh. The solution process is assumed to start on a mesh where the problem is badly…

Numerical Analysis · Mathematics 2016-02-16 Sara Pollock

We present a fourth-order finite-volume algorithm in space and time for low Mach number reacting flow with detailed kinetics and transport. Our temporal integration scheme is based on a multi-implicit spectral deferred correction (MISDC)…

Numerical Analysis · Mathematics 2016-08-24 Will Pazner , Andrew Nonaka , John Bell , Marcus Day , Michael Minion

A time-stepping L1 scheme for subdiffusion equation with a Riemann--Liouville time-fractional derivative is developed and analyzed. This is the first paper to show that the L1 scheme for the model problem under consideration is second-order…

Numerical Analysis · Mathematics 2019-09-17 Kassem Mustapha

This study presents constructions of the space-time Conservation Element and Solution Element (CESE) methods to accommodate adaptive unstructured quadrilateral meshes. Subsequently, a novel algorithm is devised to effectively manage the…

Fluid Dynamics · Physics 2025-03-10 Lisong Shi , Chaoxiong Zhang , Chih-Yung Wen

Discrete-state, continuous-time Markov models are widely used in the modeling of biochemical reaction networks. Their complexity often precludes analytic solution, and we rely on stochastic simulation algorithms to estimate system…

Quantitative Methods · Quantitative Biology 2016-05-20 Christopher Lester , Christian A. Yates , Michael B. Giles , Ruth E. Baker

To achieve efficient and accurate long-time integration, we propose a fast, accurate, and stable high-order numerical method for solving fractional-in-space reaction-diffusion equations. The proposed method is explicit in nature and…

Numerical Analysis · Mathematics 2020-03-31 Almushaira Mustafa , Harish Bhatt

We propose an adaptive numerical solver for the study of viscoelastic 2D two-phase flows using the volume-of-fluid method. The scheme uses the robust log conformation tensor technique of Fattal & Kupferman (2004,2005} combined with the…

Fluid Dynamics · Physics 2018-07-03 J. M. Lopez-Herrera , S. Popinet , A. A. Castrejon-Pita

In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate the degree to which RF automatically adapts to the intrinsic…

Machine Learning · Statistics 2026-02-24 Saptarshi Roy , Alessandro Rinaldo , Purnamrita Sarkar

Implicit time-stepping for advection is applied locally in space and time where Courant numbers are large, but standard explicit time-stepping is used for the remaining solution which is typically the majority. This adaptively implicit…

Fluid Dynamics · Physics 2024-06-14 Hilary Weller , Christian Kuehnlein , Piotr K. Smolarkiewicz

This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each…

Numerical Analysis · Mathematics 2024-04-24 Fatemeh P. A. Beik , Michele Benzi , Mehdi Najafi-Kalyani

In this paper, we consider the classical wave equation with time-dependent, spatially multiscale coefficients. We propose a fully discrete computational multiscale method in the spirit of the localized orthogonal decomposition in space with…

Numerical Analysis · Mathematics 2021-07-30 Bernhard Maier , Barbara Verfürth

Computing accurate rate constants for catalytic events occurring at the surface of a given material represents a challenging task with multiple potential applications in chemistry. To address this question, we propose an approach based on a…

The Adaptive Resolution Scheme (AdResS) is a hybrid scheme that allows one to treat a molecular system with different levels of resolution depending on the location of the molecules. The construction of a Hamiltonian based on the this idea…