# Similarities and differences of two exponential schemes for convection-diffusion problems: the FV-CF and ENATE schemes

Víctor J. Llorente (Corresponding author), Jan H.M. ten Thije Boonkkamp, Antonio Pascau, Martijn J.H. Anthonissen

### Abstract

In this paper, we present a comparison of two novel exponential schemes for convection-diffusion problems. An exponential scheme uses in one way or another the analytical solution of the flux of a one-dimensional (1D) transport equation thereby improving the results of the simulation. In a multidimensional problem, the 1D solution is combined with operator splitting. The two approximations to be assessed are the Finite Volume-Complete Flux (FV-CF) and the Enhanced Numerical Approximation of a Transport Equation (ENATE) schemes. They were proposed by the two groups that co-author the current paper. Both schemes share many similarities in 1D but differ, especially in 2D, in some aspects that will be highlighted. In their derivation the algebraic coefficients of the computational stencil are integrals of flow parameters whose calculation is crucial for the accuracy of either method. These factors and their various approximations will be analysed. Some test cases will be used to check the ability of both schemes to provide accurate results.

Original language English 124700 21 Applied Mathematics and Computation 365 https://doi.org/10.1016/j.amc.2019.124700 Published - 15 Jan 2020

### Fingerprint

Convection-diffusion Problems
Numerical Approximation
Transport Equation
Finite Volume
Fluxes
Operator Splitting
Approximation
Analytical Solution
Similarity
Convection
Coefficient
Simulation

### Keywords

• Computational Fluid Dynamics (CFD)
• ENATE
• Exponential scheme
• FV-CF
• Transport equation

### Cite this

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title = "Similarities and differences of two exponential schemes for convection-diffusion problems: the FV-CF and ENATE schemes",
abstract = "In this paper, we present a comparison of two novel exponential schemes for convection-diffusion problems. An exponential scheme uses in one way or another the analytical solution of the flux of a one-dimensional (1D) transport equation thereby improving the results of the simulation. In a multidimensional problem, the 1D solution is combined with operator splitting. The two approximations to be assessed are the Finite Volume-Complete Flux (FV-CF) and the Enhanced Numerical Approximation of a Transport Equation (ENATE) schemes. They were proposed by the two groups that co-author the current paper. Both schemes share many similarities in 1D but differ, especially in 2D, in some aspects that will be highlighted. In their derivation the algebraic coefficients of the computational stencil are integrals of flow parameters whose calculation is crucial for the accuracy of either method. These factors and their various approximations will be analysed. Some test cases will be used to check the ability of both schemes to provide accurate results.",
keywords = "Computational Fluid Dynamics (CFD), ENATE, Exponential scheme, FV-CF, Transport equation",
author = "Llorente, {V{\'i}ctor J.} and {ten Thije Boonkkamp}, {Jan H.M.} and Antonio Pascau and Anthonissen, {Martijn J.H.}",
year = "2020",
month = "1",
day = "15",
doi = "10.1016/j.amc.2019.124700",
language = "English",
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journal = "Applied Mathematics and Computation",
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Similarities and differences of two exponential schemes for convection-diffusion problems : the FV-CF and ENATE schemes. / Llorente, Víctor J. (Corresponding author); ten Thije Boonkkamp, Jan H.M.; Pascau, Antonio; Anthonissen, Martijn J.H.

In: Applied Mathematics and Computation, Vol. 365, 124700, 15.01.2020.

TY - JOUR

T1 - Similarities and differences of two exponential schemes for convection-diffusion problems

T2 - the FV-CF and ENATE schemes

AU - Llorente, Víctor J.

AU - ten Thije Boonkkamp, Jan H.M.

AU - Pascau, Antonio

AU - Anthonissen, Martijn J.H.

PY - 2020/1/15

Y1 - 2020/1/15

N2 - In this paper, we present a comparison of two novel exponential schemes for convection-diffusion problems. An exponential scheme uses in one way or another the analytical solution of the flux of a one-dimensional (1D) transport equation thereby improving the results of the simulation. In a multidimensional problem, the 1D solution is combined with operator splitting. The two approximations to be assessed are the Finite Volume-Complete Flux (FV-CF) and the Enhanced Numerical Approximation of a Transport Equation (ENATE) schemes. They were proposed by the two groups that co-author the current paper. Both schemes share many similarities in 1D but differ, especially in 2D, in some aspects that will be highlighted. In their derivation the algebraic coefficients of the computational stencil are integrals of flow parameters whose calculation is crucial for the accuracy of either method. These factors and their various approximations will be analysed. Some test cases will be used to check the ability of both schemes to provide accurate results.

AB - In this paper, we present a comparison of two novel exponential schemes for convection-diffusion problems. An exponential scheme uses in one way or another the analytical solution of the flux of a one-dimensional (1D) transport equation thereby improving the results of the simulation. In a multidimensional problem, the 1D solution is combined with operator splitting. The two approximations to be assessed are the Finite Volume-Complete Flux (FV-CF) and the Enhanced Numerical Approximation of a Transport Equation (ENATE) schemes. They were proposed by the two groups that co-author the current paper. Both schemes share many similarities in 1D but differ, especially in 2D, in some aspects that will be highlighted. In their derivation the algebraic coefficients of the computational stencil are integrals of flow parameters whose calculation is crucial for the accuracy of either method. These factors and their various approximations will be analysed. Some test cases will be used to check the ability of both schemes to provide accurate results.

KW - Computational Fluid Dynamics (CFD)

KW - ENATE

KW - Exponential scheme

KW - FV-CF

KW - Transport equation

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DO - 10.1016/j.amc.2019.124700

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