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ABSTRACT
The effect of round-off errors on the solution of numerical heat transfer is illustrated by a simple example both analytically and numerically. It is found that the upper bound of the round-off error under both conditions with or without an inner heat source is proportional to the square of grid number—n2. Increase in grid number might lead to larger round-off errors. The magnitude of relative round-off error is also determined by the specific problem. Proper treatment of the computation procedure can reduce the round-off error obviously. The precision can be improved with this method without occupation of additional computational resources.
Nomenclature
| A, B, C, P, Q | = | coefficients in TDMA |
| a, b | = | coefficients |
| Bi | = | Biot number |
| E | = | round-off error |
| Er | = | relative round-off error |
| e | = | exponent; relative error in basic operations |
| F | = | set of floating-point number |
| F | = | cross-sectional area, m2 |
| f | = | arbitrary function |
| fl | = | floating point |
| h | = | convective heat transfer coefficient, W/m2 |
| k | = | overall heat transfer coefficients, W/m2/K |
| L | = | length of the slab, m |
| M1 | = | grid number of the rightmost grid point |
| m | = | significant |
| op | = | operation |
| p | = | precision |
| R | = | set of real number |
| ro | = | round function |
| S | = | inner heat source, W/m3 |
| s | = | sign |
| T | = | temperature, °C |
| x | = | arbitrary vector |
| x | = | variable; coordinate along the slab, m |
| δ | = | maximal absolute error for numbers close to zero |
| δx | = | distance between grid points, m |
| ε | = | relative error in basic operations |
| ò | = | machine precision |
| λ | = | thermal conductivity, W/m/K |
| ϕ | = | solution of the PDE |
| Subscripts | = | |
| 1 | = | left side of the slab |
| 2 | = | right side of the slab |
| E | = | east |
| f | = | fluid |
| i | = | TDMA calculation step; grid number |
| n | = | total grid number |
| P | = | current grid point |
| up | = | upper bound |
| W | = | west |
Nomenclature
| A, B, C, P, Q | = | coefficients in TDMA |
| a, b | = | coefficients |
| Bi | = | Biot number |
| E | = | round-off error |
| Er | = | relative round-off error |
| e | = | exponent; relative error in basic operations |
| F | = | set of floating-point number |
| F | = | cross-sectional area, m2 |
| f | = | arbitrary function |
| fl | = | floating point |
| h | = | convective heat transfer coefficient, W/m2 |
| k | = | overall heat transfer coefficients, W/m2/K |
| L | = | length of the slab, m |
| M1 | = | grid number of the rightmost grid point |
| m | = | significant |
| op | = | operation |
| p | = | precision |
| R | = | set of real number |
| ro | = | round function |
| S | = | inner heat source, W/m3 |
| s | = | sign |
| T | = | temperature, °C |
| x | = | arbitrary vector |
| x | = | variable; coordinate along the slab, m |
| δ | = | maximal absolute error for numbers close to zero |
| δx | = | distance between grid points, m |
| ε | = | relative error in basic operations |
| ò | = | machine precision |
| λ | = | thermal conductivity, W/m/K |
| ϕ | = | solution of the PDE |
| Subscripts | = | |
| 1 | = | left side of the slab |
| 2 | = | right side of the slab |
| E | = | east |
| f | = | fluid |
| i | = | TDMA calculation step; grid number |
| n | = | total grid number |
| P | = | current grid point |
| up | = | upper bound |
| W | = | west |
Acknowledgment
This work has been financially supported by the National Key Research and Development Program — China (2016YFB0601201).