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Concept# Numerical methods for ordinary differential equations

Summary

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.
Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution.
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
The problem
A first-order different

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This project is developed within the scope of HydroContest which is an inter-school competition for the design of a racing boat with a high focus on energetic efficiency; the goal is to maximize the speed of the boat under the constraint of a limited power source. Hydrofoils are especially interesting since they offer an important reduction of drag at high speeds while remaining cost efficient. Within the contest, this project aims at delivering a prediction tool for the hydrofoil performance using numerical simulations of the incompressible Navier-Stokes equations approximated by the means of the Finite Element method with suitable stabilization techniques, such as the Variational Multiscale Method; we consider P1 Finite Elements with a second order BDF time discretization scheme. An automated meshing script was developed to handle arbitrary foil geometries and angles of attack. The numerical simulations were conducted using the LifeV Finite Element Library in a parallel setting. Satisfactory results have been obtained using this approach for Reynolds numbers up to 1 million.

2014In the present work, we investigate mathematical and numerical aspects of interior penalty finite element methods for free surface flows. We consider the incompressible Navier-Stokes equations with variable density and viscosity, combined with a front capturing model using the level set method. We formulate interior penalty finite element methods for both the Navier-Stokes equations and the level set advection equation. For the two-fluid Stokes equations, we propose and analyze an unfitted finite element scheme with interior penalty. Optimal a priori error estimates for the velocity and the pressure are proved in the energy norm. A preconditioning strategy with adaptive reuse of incomplete factorizations as preconditioners for Krylov subspace methods is introduced and applied for solving the linear systems. Different and complementary solutions for reducing the matrix assembly time and the memory consumption are proposed and tested, each of which is applicable in general in the context of either multiphase flow or interior penalty stabilization. As level set reinitialization method, we apply a combination of the interface local projection and a fast marching scheme. We provide for the latter a reformulation of the distance computation algorithm on unstructured simplicial meshes in any spatial dimension, allowing for both an efficient implementation and geometric insight. We present and discuss numerical solutions of reference problems for the one-fluid Navier-Stokes equations and for the level set advection problem. Solutions of benchmark problems in two and three dimensions involving one or two fluids are then approximated, and the results are compared to literature values. Finally, we describe software design techniques and abstractions for the efficient and general implementation of the applied methods.

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Mixed-precision algorithms combine low-and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a naive mixed-precision implementation of any Runge-Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed precision RKC schemes that are instead unaffected by this limiting behavior. We present three mixed-precision schemes: a first-and a second-order RKC method, and a first order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first-and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method. (C) 2022 Elsevier Inc. All rights reserved.