New explicit and implicit “improved euler” methods for the integration of ordinary differential equations

The simulation of combustion chemistry in internal combustion engines is challenging due to the need to include detailed reaction mechanisms to describe the engine physics. Computational times needed for coupling full chemistry to CFD simulations are still too computationally demanding, even when distributed computer systems are exploited. For these reasons the present paper proposes a time scale separation approach for the integration of the chemistry differential equations and applies it in an engine CFD code. The time scale separation is achieved through the estimation of a characteristic time for each of the species and the introduction of a sampling timestep, wherein the chemistry is subcycled during the overall integration. This allows explicit integration of the system to be carried out, and the step size is governed by tolerance requirements. During the subcycles each of the species is only integrated up to its own characteristic timescale, thus reducing the computational effort needed by the solver. The present ODE solver was first validated using constant pressure batch reactor simulations with two different reaction mechanisms. Then the solver was coupled with the KIVA-4 code, and validated using HCCI and DI diesel combustion cases. Performance is compared with the commonly used DVODE chemistry solver and the results show that significant reductions in the total computational time with comparable accuracy are obtained with the new solution methodology.

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“…The explicit, second-order IE integrator scheme [23] considers that a first, simple Euler integration is performed over the current timestep h:…”

Section: Explicit Integration With Time Scale Separationmentioning

confidence: 99%

An Analysis on Time Scale Separation for Engine Simulations with Detailed Chemistry

Perini

Cantore

Reitz

2011

SAE Technical Paper Series

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Section: Introductionmentioning

confidence: 99%

“…Historically, the explicit Euler method has been one of the oldest and most classical numerical methods, which is compulsory part in the textbooks and monographs, see, for example, [4,9,15,17]. It and other improved versions were frequently used as the time integrator for the numerical solutions of ordinary differential equations or partial differential equations [5,8,10,11,14,18]. However, most of the time we ignore a fact that explicit Euler method could generate superconvergence with a specific step-ratio when it is applied to solve the convection-diffusion problems [19].…”

Section: Introductionmentioning

confidence: 99%

Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases

Zhang

Sun

2023

Numerical Methods Partial

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This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high‐dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal derivative and centered difference discretization in spatial derivatives. The priori estimate of the improved difference scheme with application to the constant convection coefficients is performed under the maximum norm and the optimal convergence rate four is achieved when the step‐ratios along each direction equal to . Also we give partial results for the three‐dimensional case. The improved difference schemes have essentially improved the CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two‐/three‐dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee–Infante equation and the Burgers' equation substantiate the good properties claimed for the improved difference scheme.

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“…Historically, the explicit Euler method has been one of the most classical and oldest numerical methods, which is compulsory part in our textbooks and monographs, see e.g., [3,7,12,13]. It and other improved versions were frequently used as the time integrator for the numerical solutions of ordinary differential equations or partial differential equations [4,6,8,11,14]. However, most of the time we ignore the fact that explicit Euler method could generate superconvergence with a specific step-ratio when it is applied to solve the convection-diffusion problems [15].…”

Section: Introductionmentioning

confidence: 99%

“…Throughout the whole paper, we always assume that the exact solutions to (1.1) or (1.2) are sufficiently smooth in the sense that u(x, t) ∈ C 6,6,4 ( Ω × [0, T ]) for two-dimensional problem and u(x, t) ∈ C 6,6,6,4 ( Ω × [0, T ]) for three-dimensional problem.…”

Section: Introductionmentioning

confidence: 99%

Optimal convergence rate of the explicit Euler method for convection-diffusion equations II: high dimensional cases

Zhang¹,

Zhang²,

Sun³

2022

Preprint

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This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection-diffusion equations [Appl. Math. Lett. 131 (2022) 108048] which focuses on high-dimensional linear/nonlinear cases under Dirichlet or Neumann boundary conditions. Several new corrected difference schemes are proposed based on the explicit Euler discretization in temporal derivative and central difference discretization in spatial derivatives. The priori estimate of the corrected scheme with application to constant convection coefficients is provided at length by the maximum principle and the optimal convergence rate four is proved when the step ratios along each direction equal to 1/6. The corrected difference schemes have essentially improved CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two-/three-dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee-Infante equation, the Burgers' equation and classification to name a few substantiate the good properties claimed for the corrected difference scheme.

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New explicit and implicit “improved euler” methods for the integration of ordinary differential equations

Cited by 12 publications

References 0 publications

An Analysis on Time Scale Separation for Engine Simulations with Detailed Chemistry

An Analysis on Time Scale Separation for Engine Simulations with Detailed Chemistry

Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases

Optimal convergence rate of the explicit Euler method for convection-diffusion equations II: high dimensional cases

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