A Survey of the Implementation of Numerical Schemes for the Heat Equation Using Forward Euler in Time
Author(s)
Pedro Pablo Cárdenas Alzate*
ABSTRACT
We establish the conditions for the compute of the Global Truncation Error (GTE), stability restriction on the time step and we prove the consistency using forward Euler in time and a fourth order discretization in space for Heat Equation with smooth initial conditions and Dirichlet boundary conditions.
We establish the conditions for the compute of the Global Truncation Error (GTE), stability restriction on the time step and we prove the consistency using forward Euler in time and a fourth order discretization in space for Heat Equation with smooth initial conditions and Dirichlet boundary conditions.
Cite this paper
Alzate, P. (2014) A Survey of the Implementation of Numerical Schemes for the Heat Equation Using Forward Euler in Time. Journal of Applied Mathematics and Physics, 2, 1153-1158. doi: 10.4236/jamp.2014.213135.
Alzate, P. (2014) A Survey of the Implementation of Numerical Schemes for the Heat Equation Using Forward Euler in Time. Journal of Applied Mathematics and Physics, 2, 1153-1158. doi: 10.4236/jamp.2014.213135.
References
[1] Strikwerda, J.C. (1989) Finite Difference Schemes and Partial Differential Equations. Wadsworth & Brooks, USA. [2] McRea, G.J. and Godin, W.R. (1967) Numerical Solution of Atmospheric Diffusion for Chemically Reacting Flows. Journal of Computational Physics, 77, 1-42. [3] Hundsdorfer, W. and Koren, B. (1995) A Positive Finite-Difference Advection Scheme Applied on Locally Refined Grids. Journal of Computational Physics, 117, 35-36.
http://dx.doi.org/10.1006/jcph.1995.1042 [4] Morton, K.W. (1980) Stability of Finite Difference Approximations to a Diffusion-Convection Equation. International Journal for Numerical Methods Engineering, 15, 677-683.
http://dx.doi.org/10.1002/nme.1620150505 [5] Canuto, C. and Hussaini, M. (1988) Spectral Methods in Fluids Dynamics. Springer Series in Computational Physics, Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-84108-8 [6] Dehghan, M. (2005) On the Numerical Solution of the One-Dimensional Convection-Diffusion Equation. Mathematical Problems in Engineering, 1, 61-74. [7] Dehghan, M. (2007) The One-Dimensional Heat Equation Subject to a Boundary Integral Specification. Chaos, Solitons & Fractals, 32, 661-675.
http://dx.doi.org/10.1155/MPE.2005.61 [8] Mitchell, A.R. and Griffiths, D.F. (1980) The Finite Difference Method in Partial Differential Equations. John Wiley & Sons, Chichester. [9] Mickens, R.E. (2000) Applications of Nonstandard Finite Differences Schemes. World Scientific Publishing, River Edge. [10] Lu, X., et al. (2005) A New Analytical Method to Solve the Heat Equation for a Multi-Dimensional Composite Slab. Journal of Physics, 38, 2873.
http://doi:10.1088/0305-4470/38/13/004
[1] Strikwerda, J.C. (1989) Finite Difference Schemes and Partial Differential Equations. Wadsworth & Brooks, USA. [2] McRea, G.J. and Godin, W.R. (1967) Numerical Solution of Atmospheric Diffusion for Chemically Reacting Flows. Journal of Computational Physics, 77, 1-42. [3] Hundsdorfer, W. and Koren, B. (1995) A Positive Finite-Difference Advection Scheme Applied on Locally Refined Grids. Journal of Computational Physics, 117, 35-36.
http://dx.doi.org/10.1006/jcph.1995.1042 [4] Morton, K.W. (1980) Stability of Finite Difference Approximations to a Diffusion-Convection Equation. International Journal for Numerical Methods Engineering, 15, 677-683.
http://dx.doi.org/10.1002/nme.1620150505 [5] Canuto, C. and Hussaini, M. (1988) Spectral Methods in Fluids Dynamics. Springer Series in Computational Physics, Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-84108-8 [6] Dehghan, M. (2005) On the Numerical Solution of the One-Dimensional Convection-Diffusion Equation. Mathematical Problems in Engineering, 1, 61-74. [7] Dehghan, M. (2007) The One-Dimensional Heat Equation Subject to a Boundary Integral Specification. Chaos, Solitons & Fractals, 32, 661-675.
http://dx.doi.org/10.1155/MPE.2005.61 [8] Mitchell, A.R. and Griffiths, D.F. (1980) The Finite Difference Method in Partial Differential Equations. John Wiley & Sons, Chichester. [9] Mickens, R.E. (2000) Applications of Nonstandard Finite Differences Schemes. World Scientific Publishing, River Edge. [10] Lu, X., et al. (2005) A New Analytical Method to Solve the Heat Equation for a Multi-Dimensional Composite Slab. Journal of Physics, 38, 2873.
http://doi:10.1088/0305-4470/38/13/004