Finite Difference Methods for Weather Prediction in Abuja, Nigeria

International Journal of Scientific Research in Computer Science, Engineering and Information Technology © 2020 IJSRCSEIT | Volume 6 | Issue 3 | ISSN : 2456-3307 DOI : https://doi.org/10.32628/CSEIT2063207 Finite Difference Methods for Weather Prediction in Abuja, Nigeria Jacob Emmanuel1, Ogunfiditimi F.O.2, Victor Alexander Okhuese3, Odeyemi J.K4 1 Department of Mathematics, University of Abuja, Abuja, Nigeria 2 Department of Mathematics, University of Abuja, Abuja, Nigeria 3 College of Pure and Applied Sciences, Jomo Kenyatta University of Agriculture and Technology, Kenya. 4 Department of Mathematics, University of Abuja, Abuja, Nigeria Corresponding Author : [email protected] ABSTRACT In this study, we simulate some finite difference schemes numerically to predict weather trends of Abuja Station, Nigeria. By evaluating the results from these schemes, it has shown that the best scheme in the finite difference method that gives a close accurate weather forecast is the trapezoidal scheme hence we use it to simulate numerical weather data obtained from Federal Airports Authority of Nigeria (FAAN), Abuja and corresponding numerical weather data obtained by the compatible finite difference schemes, using MATLAB (R2012a) software to obtain future numerical weather trends. Keywords : finite different, trapezoidal scheme, Weather Prediction, forecasting, modelling I. INTRODUCTION to predict the weather and future sea state (the process of numerical weather prediction) has changed over Weather forecasting is one of the most complex and the years. Though first attempted in the 1920s, it was remarkably problems of modern science. In spite of not until the advent of the computer and computer evident advancement in the few decades and shift simulation that computation time was reduced to less from manual forecasting methods to numerical ones, than the forecast period itself. there are some significant problems that are yet to be solved either by manual methods or methods based on However, the vast range of available finite difference computer simulation posing interesting challenges for scheme is both a blessing and a curse, and many all those engaged in the field. An interminably need different combinations have been proposed, analysed for in depth information on the actual meteorological and used for large scale geophysical fluid dynamics conditions and problems associated to the use of applications, particularly in the ocean modeling traditional methods are responsible from intensive community Le Roux et al. (2005, 2007); Le Roux and development of numerical weather prediction (NWP). Pouliot (2008); Danilov (2010); Cotter et al. (2009); Cotter and Ham (2011); Rostand and Le Roux (2008); The history of numerical weather prediction considers Le Roux (2012); Comblen et al. (2010), whilst many how current weather conditions as input into other combinations have been used in engineering mathematical models of the atmosphere and oceans Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 213 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 applications where different scales and modeling by reference to the advection equation which we write aspects are important. in the form In geophysical applications, the stability properties of compatible finite difference have long been recognized, leading to various choices being proposed and analyzed on triangular meshes, Walters and Casulli (1998); Rostand and Le Roux (2008). However, no explicit use was made of the compatible structure beyond stability until Cotter and Shipton (2012) used it, which proved that all compatible finite difference methods have exactly steady geotropic modes; this is (2) 𝜕𝑢 𝜕𝑡 𝜕𝑢 + 𝑐 𝜕𝑥 = 0 where 𝑐 is a constant. We divide the (𝑥, 𝑡) −plane into a series of discrete points (𝑖∆𝑥, 𝑛∆𝑡) and denote the approximate solution for u at this point by 𝑢𝑖𝑛 . The possible finite-difference scheme for the equation is 𝑢𝑖𝑛+1 −𝑢𝑖𝑛 ∆𝑡 +𝑐 𝑛 𝑢𝑖𝑛 −𝑢𝑖−1 ∆𝑥 =0 (3) We may rewrite (3) as 𝑛 𝑢𝑖𝑛+1 = (1 − 𝜇)𝑢𝑖𝑛 + 𝜇𝑢𝑖−1 , (4) prediction Staniforth and Thuburn (2012). where µ = 𝑐∆𝑡/∆𝑥. The advection equation Eq. (2) has The purpose of this paper is to obtain numerical equation in the form of a single harmonic is considered a crucial property for numerical weather a possible finite-difference scheme given by Eq. (3) and hence an analytic solution of the advection weather data and weather trends using the various schemes of the Finite Difference Method because the Finite Difference Method is one of the most powerful 𝑢(𝑥, 𝑡) = 𝑅𝑒[𝑈(𝑡)𝑒 𝑖𝑘𝑥 ] (5) Here 𝑈(𝑡) is the wave amplitude and 𝑘 the wavenumber. Substituting this result into Eq. (2) gives 𝑑𝑈 𝑑𝑡 numerical methods for obtaining the numerical + 𝑖𝑘𝑐𝑈 = 0, (6) which has the solution solution of step-wise differential equations. (7) Finite Difference Approximations for Finite Difference Methods The finite difference method involves using discrete 𝑈(𝑡) = 𝑈(0)𝑒 −𝑖𝑘𝑐𝑡 , 𝑈(0) which is the initial amplitude. Hence 𝑢(𝑥, 𝑡) = 𝑅𝑒[𝑈(0)𝑒 −𝑖𝑘(𝑥−𝑐𝑡) ] approximations like (8) as expected. The solution is finally expressed in Eq. (8). 𝜕𝑢 𝜕𝑥 ≈ 𝑢𝑖+1 −𝑢𝑖 Δ𝑥 (1) where the quantities on the right hand side are defined However, in the von Neumann method we looked for an analogous solution of the finite-difference equation Eq. (4) which after substituting 𝑢𝑗𝑛 = 𝑅𝑒[𝑈 (𝑛) 𝑒 𝑖𝑘𝑗∆𝑥 ], on the finite difference mesh. Approximations to the governing differential equation are obtained by this reduces the entire scheme to the amplitude replacing all continuous derivatives by discrete equation; formulas such as those in Eq. (1). Advection Equation The advection equation is the major model used in this weather prediction meanwhile other schemes were derived based on their stability, conditional stability and neutrality as it affect the weather trends in a local station. Many of the important ideas can be illustrated Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 𝑈 (𝑛+1) = 𝜆𝑈 (𝑛) (9) which properly defines the amplification factor |𝜆| and hence we can now study the behavior of the amplitude 𝑈 (𝑛) as 𝑛 increases, the stability of the scheme and the frequency of the stability is given by; 𝑝 = 𝜔∆𝑡 ∆𝑡 ≤ (10) 1 |𝜔| (11) 916 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 where 𝑝 is the stability of the scheme, 𝜆 is the wavelength 𝜔 is the frequency and ∆𝑡 the time interval and 𝜔 = 1,2, … , 𝑛. For Euler Scheme 1 |𝜆| = (1 + 𝑝2 )2 . 𝜆 = 1 + 𝑖𝑝, at 𝑝 = 1, we have (12) 1 4 (1+ 𝑖𝑝) (1+𝑝2 ) at 𝑝 = 1, we have 1 |𝜆| = (1 + 𝑝2 )−2 , 𝜆=𝑖 This scheme is stable, if |𝑝| ≤ 1. For Heun Scheme 1 2 𝑝 2 at 𝑝 = 1, we have For Backward Scheme (13) 𝜆= 1 (1+ 𝑝2 ) 4 at 𝑝 = 1, we have , unstable. itself for the resultant solution of 𝑈 (𝑛+1) = 𝜆𝑈 (𝑛) as |𝜆| = 1. 𝑈 (𝑛+1) = 𝑅𝑒[𝜆𝑈 (𝑛) ]. (17) 𝜆 = 1 + 𝑖/1.25 This scheme is always neutral. For Matsuno Scheme (16) 𝜆 = 0.5 + 𝑖 This is always > 1 so that the Heun scheme is always For Trapezoidal Scheme 1 4 + 𝑖𝑝, |𝜆| = (1 + 1 1 4 2 𝑝 ). 4 However, we select the real part minus the product of the imaginary part of the deduced wavelength with 𝜆 = 0.5 + 0.125𝑖 This scheme is stable (1+ 𝑝2 +𝑖𝑝) (15) at 𝑝 = 1, we have 𝜆 =1− 𝜆 =1+𝑖 This scheme is unstable|𝜆| > 1 for any 𝑝 > 0 𝜆= 1 𝜆 = 1 − 𝑝2 + 𝑖𝑝, |𝜆| = (1 − 𝑝2 + 𝑝4 )2 II. NUMERICAL SOLUTIONS Summary of Weather Data Set from Federal Airport Authority of Nigeria, Abuja Station Table 1 : Dataset from the Federal Airport Authority of Nigeria for Abuja Station Annual Climatological Summary Year: 2015 Station: ABUJA, NG Elev: 343.1ft. Lat: 09.15oN Lon: 07.00oE STATION # STATION NAME ELEV LAT LONG DATE RelHum TMAX TMIN RAINFALL SUNSHINE HRS WIND SPEED WIND DIR. 65125 Abuja 343.1 09.15’N 07.00’E 201501 43 35.5 19.3 0 7.3 2.9 N 65125 Abuja 343.1 09.15’N 07.00’E 201502 50 37.4 23.2 0.6 7.5 3.7 NE 65125 Abuja 343.1 09.15’N 07.00’E 201503 62 37.7 25 7.5 8.2 3.5 NE 65125 Abuja 343.1 09.15’N 07.00’E 201504 62 36.6 25.7 74.2 7.5 5 E 65125 Abuja 343.1 09.15’N 07.00’E 201505 76 35.8 24.6 109.2 7.4 4.9 SW 65125 Abuja 343.1 09.15’N 07.00’E 201506 81 30.2 23.2 267.2 7.5 4.7 S 65125 Abuja 343.1 09.15’N 07.00’E 201507 86 28.7 22.3 314.8 4.5 3.7 SW 65125 Abuja 343.1 09.15’N 07.00’E 201508 87 28.7 22.5 278.3 5.2 4.2 NW 65125 Abuja 343.1 09.15’N 07.00’E 201509 83 29.5 22.2 258.4 5.2 4.1 W 65125 Abuja 343.1 09.15’N 07.00’E 201510 78 30 21.8 238.2 6.8 3.3 NW 65125 Abuja 343.1 09.15’N 07.00’E 201511 64 33.7 21.6 Trace 9.2 3 E 65125 Abuja 343.1 09.15’N 07.00’E 201512 36 35 17.2 0 8.8 3.2 NE Source: FAAN Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 917 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 Solution of Sunshine Hours Prediction Using Finite Difference Scheme Using Eq. (17) and sunshine hours value from Table 1 for the first month, we compute the predicted values for the different schemes. 𝑈 (𝑛+1) = 𝑅𝑒[𝜆𝑈 (𝑛) ] For Heun Scheme where 𝜆 = 0.5 + 𝑖 and 𝑈 (𝑛) = 7.3 for sunshine hours then 𝑈 (𝑛+1) = 𝑅𝑒[7.3(0.5 + 𝑖)] = 𝑅𝑒[3.65 + 7.3𝑖] = 3.65 − 1 = 2.65 For Matsuno Scheme where 𝜆 = 𝑖 and 𝑈 (𝑛) = 7.3 for sunshine hours then 𝑈 (𝑛+1) = 𝑅𝑒[7.3(𝑖)] = 𝑅𝑒[7.3𝑖] = 0 For Trapezoidal Scheme where 𝜆 = 1 + 𝑖/1.25 and 𝑈 (𝑛) = 7.3 for sunshine hours then 𝑈 (𝑛+1) = 𝑅𝑒[7.3(1 + 𝑖/1.25)] = 7.66 − 1 = 6.66 For Backward Scheme where 𝜆 = 0.5 + 0.125𝑖 and 𝑈 (𝑛) = 7.3 for sunshine hours then 𝑈 (𝑛+1) = 𝑅𝑒[7.3(0.5 + 0.125𝑖)] = 4.6 − 1 = 3.6 For Euler Scheme where 𝜆 = 1 + 𝑖 and 𝑈 (𝑛) = 7.3 for sunshine hours then 𝑈 (𝑛+1) = 𝑅𝑒[7.3(1 + 𝑖)] = 7.3 − 1 = 6.3 The results of predicted sunshine hours for all the 12 months of the year are shown in Table 2 Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 205 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 Table 2 : Sunshine Hours 2016 (SHRs 2016) Months 1 2 3 4 5 6 7 8 9 10 11 12 𝜔 Wavelength 𝜆 Heun Matsuno Trapezoidal Backward 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 𝑖 1 + 𝑖/1.25 𝑖 1 + 𝑖/1.25 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 Amplitude 𝑈 (𝑛) Euler Sunshine 2015 1 + 𝑖 7.3 Schemes 𝑈 (𝑛+1) (SHRs 2016) Heun Matsuno Trapezoidal Backward Euler 2.65 0 6.66 3.6 6.3 1+𝑖 7.5 2.75 0 6.86 3.7 6.5 1+𝑖 8.2 3.1 0 7.56 4.08 7.2 1+𝑖 7.5 2.75 0 6.86 3.7 6.5 1+𝑖 7.4 2.7 0 6.76 3.68 6.4 1+𝑖 7.5 2.75 0 6.86 3.7 6.5 1+𝑖 4.5 1.25 0 3.86 2.2 3.5 1+𝑖 5.2 1.6 0 4.56 2.58 4.2 1+𝑖 5.2 1.6 0 4.56 2.58 4.2 1+𝑖 6.8 2.4 0 6.16 3.38 5.8 1+𝑖 9.2 3.6 0 8.56 4.58 8.2 1+𝑖 8.8 3.4 0 8.16 4.38 7.8 From the above table, the selection of the scheme to represent the model forecasting for the sunshine hours for 2016 is based on the trend of the scheme whose result is closest to the previous year i.e. 2015 and hence among all five schemes in the table it is very obvious that aside Euler’s (Forward) Scheme which is the second closest, the Trapezoidal Scheme is the closest to the given sunshine hours in 2015. Hence we use the Sunshine Hours predicted using the Trapezoidal Scheme. Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 919 International Journal of Scientific Research in Computer Science, Engineering and Information Technology © 2020 IJSRCSEIT | Volume 6 | Issue 3 | ISSN : 2456-3307 DOI : https://doi.org/10.32628/CSEIT2063207 A comparative Chart Showing the Sunshine Hours Deduced by Various Scheme 10 9 Sunshine Hours for the Various Schemes 8 7 SunShine Heun Trapezodial Backward Euler 6 5 4 3 2 1 0 2 4 6 Months of the Year 8 10 12 Figure 1 : A Comparative Chart Showing the Sunshine Hours Deduced by Various Schemes in one year Observing our choice Trapezoidal Scheme from Figure 1 above it is obviously showing that the sunshine hours between January and May will be relatively high and will begin to decrease from June and start to rise again around September and falls again in December. Solution of Wind Speed Prediction Using Finite Difference Scheme Using equation (17) and wind speed value from Table 1 for the third month, we compute the predicted values for the different schemes. 𝑈 (𝑛+1) = 𝑅𝑒[𝜆𝑈 (𝑛) ] For Heun Scheme where 𝜆 = 0.5 + 𝑖 and 𝑈 (𝑛) = 3.5 for wind speed then For Matsuno Scheme 𝑈 (𝑛+1) = 𝑅𝑒[3.5(0.5 + 𝑖)] = 𝑅𝑒[1.75 + 3.5𝑖] = 1.75 − 1 = 0.75 where 𝜆 = 𝑖 and 𝑈 (𝑛) = 3.5 for wind speed then For Trapezoidal Scheme 𝑈 (𝑛+1) = 𝑅𝑒[3.5(𝑖)] = 𝑅𝑒[3.5𝑖] = 0 where 𝜆 = 1 + 𝑖/1.25 and 𝑈 (𝑛) = 3.5 for wind speed then 𝑈 (𝑛+1) = 𝑅𝑒[3.5(1 + 𝑖/1.25)] = 3.86 − 1 = 2.86 Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 919 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 For Backward Scheme where 𝜆 = 0.5 + 0.125𝑖 and 𝑈 (𝑛) = 3.5 for wind speed then 𝑈 (𝑛+1) = 𝑅𝑒[3.5(0.5 + 0.125𝑖)] = 2.7 − 1 = 1.7 For Euler Scheme where 𝜆 = 1 + 𝑖 and 𝑈 (𝑛) = 7.3 for wind speed then 𝑈 (𝑛+1) = 𝑅𝑒[3.5(1 + 𝑖)] = 3.5 − 1 = 2.5 The results of predicted wind speed for all the 12 months of the year are shown in Table 3 Table 3 : Wind Speed 2016 (WS 2016) Months 1 2 3 4 5 6 7 8 9 10 11 12 𝜔 Wavelength 𝜆 Heun Matsuno Trapezoidal Backward 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 𝑖 1 + 𝑖/1.25 𝑖 1 + 𝑖/1.25 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 Amplitude 𝑈 (𝑛) Euler Wind Speed’15 1 + 𝑖 2.9 Schemes 𝑈 (𝑛+1) (WS 2016) Heun Matsuno Trapezoidal Backward Euler 0.45 0 2.26 1.4 1.9 1+𝑖 3.7 0.85 0 3.06 1.8 2.7 1+𝑖 3.5 0.75 0 2.86 1.7 2.5 1+𝑖 5 1.5 0 4.36 2.48 4 1+𝑖 4.9 1.45 0 4.26 2.4 3.9 1+𝑖 4.7 1.35 0 4.06 2.3 3.7 1+𝑖 3.7 0.85 0 3.06 1.8 2.7 1+𝑖 4.2 1.1 0 3.56 2.08 3.2 1+𝑖 4.1 1.05 0 3.46 2 3.1 1+𝑖 3.3 0.65 0 2.66 1.49 2.3 1+𝑖 3 0.5 0 2.36 1.48 2 1+𝑖 3.2 0.6 0 2.56 1.58 2.2 From the above table, the selection of the scheme to represent the model forecasting for the Wind Speed for 2016 is based on the trend of the scheme whose result is closest to the previous year i.e. 2015 and hence among all five schemes in the table it is very obvious that aside Euler’s (Forward) Scheme which is the second most closest, the Trapezoidal Scheme is the most closest to the given Wind Speed in 2015. Hence we use the Wind Speed predicted using the Trapezoidal Scheme. Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 920 International Journal of Scientific Research in Computer Science, Engineering and Information Technology © 2020 IJSRCSEIT | Volume 6 | Issue 3 | ISSN : 2456-3307 DOI : https://doi.org/10.32628/CSEIT2063207 A comparative Chart Showing the Wind Speed Deduced by Various Scheme 5 Wind Speed Heun Trapezodial Backward Euler 4.5 Wind Speed for the Various Schemes 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 6 Months of the Year 8 10 12 Figure 2: A Comparative Chart Showing the Wind Speed Deduced by Various Schemes in one year Observing our choice Trapezoidal Scheme from Figure 2 above it is obviously showing that the wind speed will increase from January to May and will begin to decrease from June and start to rise again around September and falls again in November. Solution of Rainfall Prediction Using Finite Difference Scheme Using equation (17) and rainfall value from Table 1 for the second month, we compute the predicted values for the different schemes. 𝑈 (𝑛+1) = 𝑅𝑒[𝜆𝑈 (𝑛) ] For Heun Scheme where 𝜆 = 0.5 + 𝑖 and 𝑈 (𝑛) = 0.6 for rainfall then 𝑈 (𝑛+1) = 𝑅𝑒[0.6(0.5 + 𝑖)] = 𝑅𝑒[0.3 + 0.6𝑖] = 0.3 − 1 = −0.7 For Matsuno Scheme where 𝜆 = 𝑖 and 𝑈 (𝑛) = 0.6 for rainfall then 𝑈 (𝑛+1) = 𝑅𝑒[0.6(𝑖)] = 𝑅𝑒[0.6𝑖] = 0 For Trapezoidal Scheme where 𝜆 = 1 + 𝑖/1.25 and 𝑈 (𝑛) = 0.6 for rainfall then 𝑈 (𝑛+1) = 𝑅𝑒[0.6(1 + 𝑖/1.25)] = 0.36 − 0.8 = −0.04 For Backward Scheme where 𝜆 = 0.5 + 0.125𝑖 and 𝑈 (𝑛) = 0.6 for rainfall then Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 921 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 𝑈 (𝑛+1) = 𝑅𝑒[0.6(0.5 + 0.125𝑖)] = 0.3 − 0.015628 = 0.28 For Euler Scheme where 𝜆 = 1 + 𝑖 and 𝑈 (𝑛) = 0.6 for rainfall then 𝑈 (𝑛+1) = 𝑅𝑒[0.6(1 + 𝑖)] = 0.6 − 1 = −0.4 The results of predicted rainfall for all the 12 months of the year are shown in Table 4 Table 4: RainFall 2016 (RF 2016) Month s 𝜔 1 2 3 4 5 6 7 8 9 10 11 12 Wavelength 𝜆 Heu n 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 Matsun o 𝑖 Trapezoida l 1 + 𝑖/1.25 𝑖 1 + 𝑖/1.25 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 Backwar d 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 Schemes 𝑈 (𝑛+1) (RF 2016) Amplitude 𝑈 (𝑛) Rain Fall ‘15 0 Heun 1+𝑖 Eule r 1+𝑖 Trapezoida l -0.64 Backwar d -0.02 Euler -1 Matsun o 0 0.6 -0.7 0 -0.04 0.28 -0.4 1+𝑖 7.5 2.75 0 6.86 3.7 6.5 1+𝑖 74.2 36.1 0 73.56 37 73.2 1+𝑖 109.2 53.6 0 108.56 54.58 108.2 1+𝑖 267.2 132.6 0 266.56 133.58 266.2 1+𝑖 314.8 156.4 0 314.16 157.38 313.8 1+𝑖 278.3 0 277.66 139 277.2 1+𝑖 258.4 138.1 5 128.2 0 257.76 129 257.4 1+𝑖 238.2 118.1 0 237.56 119 237.2 1+𝑖 Trace Trace Trace Trace Trace 1+𝑖 0 -1 0 -0.64 -0.02 Trac e -1 -1 From the above table, the selection of the scheme to represent the model forecasting for the Rain Fall for 2016 is based on the trend of the scheme whose result is closest to the previous year i.e. 2015 and hence among all five schemes in the table it is very obvious that aside Euler’s (Forward) Scheme which is the second most closest, the Trapezoidal Scheme is the most closest to the given Rain Fall in 2015. Hence we use the Rain Fall predicted using the Trapezoidal Scheme. Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 922 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 A Comparative Chart Showing the Rain Fall Deduced by Various Scheme 350 Rain Fall Heun Trapezodial Backward Euler 300 Rain Fall for the Various Schemes 250 200 150 100 50 0 -50 0 2 4 6 Months of the Year 8 10 12 Figure 3 : A Comparative Chart Showing the Rain Fall Deduced by Various Schemes in one year Observing our choice Trapezoidal Scheme from Figure 3 above it is obviously showing that the rain fall will start around late February and be very high in June till around October then will begin to reduce and dry season will set in from November. 3.5 Solution of Relative Humidity Prediction Using Finite Difference Scheme Using equation (17) and relative humidity value from Table 1 for the fourth month, we compute the predicted values for the different schemes. 𝑈 (𝑛+1) = 𝑅𝑒[𝜆𝑈 (𝑛) ] For Heun Scheme where 𝜆 = 0.5 + 𝑖 and 𝑈 (𝑛) = 62 for relative humidity then 𝑈 (𝑛+1) = 𝑅𝑒[62(0.5 + 𝑖)] = 𝑅𝑒[31 + 62𝑖] = 31 − 1 = 30 For Matsuno Scheme where 𝜆 = 𝑖 and 𝑈 (𝑛) = 62 for relative humidity then 𝑈 (𝑛+1) = 𝑅𝑒[62(𝑖)] = 𝑅𝑒[62𝑖] = 0 For Trapezoidal Scheme where 𝜆 = 1 + 𝑖/1.25 and 𝑈 (𝑛) = 62for relative humidity then Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 923 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 𝑈 (𝑛+1) = 𝑅𝑒[62(1 + 𝑖/1.25)] = 62 − 0.64 = 61.36 For Backward Scheme where 𝜆 = 0.5 + 0.125𝑖 and 𝑈 (𝑛) = 62 for relative humidity then 𝑈 (𝑛+1) = 𝑅𝑒[62(0.5 + 0.125𝑖)] = 31 − 0.015625 = 30.98 For Euler Scheme where 𝜆 = 1 + 𝑖 and 𝑈 (𝑛) = 62 for relative humidity then 𝑈 (𝑛+1) = 𝑅𝑒[62(1 + 𝑖)] = 62 − 1 = 61 The results of predicted relative humidity for all the 12 months of the year are shown in Table 5 Table 5 : Relative Humidity 2016 (RH 2016) Months 𝜔 1 2 3 4 5 6 7 8 9 10 11 12 Wavelength 𝜆 Heun Matsuno 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 Amplitude 𝑈 (𝑛) Rel. Heun Hum‘15 43 20.5 Schemes 𝑈 (𝑛+1) (RH 2016) Matsuno Trapezoidal Backward Euler 0 42.36 21.48 42 1+𝑖 50 24 0 49.36 24.98 49 1+𝑖 62 30 0 61.36 30.98 61 1+𝑖 62 30 0 61.36 30.98 61 1+𝑖 76 37 0 75.36 37.98 75 1+𝑖 81 39.5 0 80.36 40.48 80 1+𝑖 86 42 0 85.36 42.98 85 1+𝑖 87 42.5 0 86.36 43.48 86 1+𝑖 83 40.5 0 82.36 41.48 82 1+𝑖 78 38 0 77.36 38.98 77 1+𝑖 64 31 0 63.36 31.98 63 1+𝑖 36 17 0 35.36 17.98 35 Trapezoidal Backward Euler 𝑖 1 + 𝑖/1.25 1+𝑖 𝑖 1 + 𝑖/1.25 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 From the above table, the selection of the scheme to represent the model forecasting for the Relative Humidity for 2016 is based on the trend of the scheme whose result is closest to the previous year i.e. 2015 and hence among all five schemes in the table it is very obvious that aside Euler’s (Forward) Scheme which is the second most closest, the Trapezoidal Scheme is the most closest to the given Relative Humidity in 2015. Hence we use the Relative Humidity predicted using the Trapezoidal Scheme. Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 924 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 A Comparative Chart Showing the Relative Humidity Deduced by Various Scheme 90 Relative Humidity Heun Trapezodial Backward Euler 80 Relative Humidity for the Various Schemes 70 60 50 40 30 20 10 0 2 4 6 Months of the Year 8 10 12 Figure 4 : A Comparative Chart Showing the Relative Humidity Deduced by Various Schemes in one year As the rain fall increase so does the relative humidity, therefore, observing our choice Trapezoidal Scheme from Figure 4 above it is obviously showing that the relative humidity will start around late February and be very high in June till around October then will begin to reduce and dry season will set in from November. Solution of Maximum Temperature Prediction Using Finite Difference Scheme Using equation (17) and maximum temperature value from Table 1 for the fifth month, we compute the predicted values for the different schemes. 𝑈 (𝑛+1) = 𝑅𝑒[𝜆𝑈 (𝑛) ] For Heun Scheme where 𝜆 = 0.5 + 𝑖 and 𝑈 (𝑛) = 35.8 for maximum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[35.8(0.5 + 𝑖)] = 𝑅𝑒[17.9 + 35.8𝑖] = 17.9 − 1 = 16.9 For Matsuno Scheme where 𝜆 = 𝑖 and 𝑈 (𝑛) = 35.8 for maximum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[35.8(𝑖)] = 𝑅𝑒[35.8𝑖] = 0 For Trapezoidal Scheme where 𝜆 = 1 + 𝑖/1.25 and 𝑈 (𝑛) = 35.8 for maximum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[35.8(1 + 𝑖/1.25)] = 35.8 − 0.64 = 35.16 Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 925 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 For Backward Scheme where 𝜆 = 0.5 + 0.125𝑖 and 𝑈 (𝑛) = 35.8 for maximum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[35.8(0.5 + 0.125𝑖)] = 17.9 − 0.015625 = 17.88 For Euler Scheme where 𝜆 = 1 + 𝑖 and 𝑈 (𝑛) = 35.8 for maximum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[35.8(1 + 𝑖)] = 35.8 − 1 = 34.8 The results of predicted maximum temperature for all the 12 months of the year are shown in Table 6 Table 6: Maximum Temperature 2016 (TMax 2016) Months 1 2 3 4 5 6 7 8 9 10 11 12 𝜔 Wavelength 𝜆 Heun 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 Matsuno 𝑖 Trapezoidal 1 + 𝑖/1.25 𝑖 1 + 𝑖/1.25 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 Backward 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 Schemes 𝑈 (𝑛+1) (TMax 2016) Amplitude 𝑈 (𝑛) TMax‘15 35.5 Heun 16.75 Matsuno 0 Trapezoidal 34.86 Backward 17.7 Euler 34.5 1+𝑖 37.4 17.7 0 36.76 18.68 36.4 1+𝑖 37.7 17.85 0 37.06 18.8 36.7 1+𝑖 36.6 17.3 0 35.96 18.3 35.6 1+𝑖 35.8 16.9 0 35.16 17.88 34.8 1+𝑖 30.2 14.1 0 29.57 15 29.2 1+𝑖 28.7 13.35 0 28.06 14.33 27.7 1+𝑖 28.7 13.35 0 28.06 14.33 27.7 1+𝑖 29.5 13.75 0 28.86 14.7 28.5 1+𝑖 30 14 0 29.36 14.98 29 1+𝑖 33.7 15.85 0 33.06 16.8 32.7 1+𝑖 35 16.5 0 34.36 17.5 34 Euler 1+𝑖 From the above table, the selection of the scheme to represent the model forecasting for the Maximum Temperature for 2016 is based on the trend of the scheme whose result is closest to the previous year i.e. 2015 and hence among all five schemes in the table it is very obvious that aside Euler’s (Forward) Scheme which is the second most closest, the Trapezoidal Scheme is the most closest to the given Maximum Temperature in 2015. Hence we use the Maximum Temperature predicted using the Trapezoidal Scheme. Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 926 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 A Comparative Chart Showing the Maximum Temperative Deduced by Various Scheme 40 Maximum Temperative for the Various Schemes 35 30 Max Temp. Heun Trapezodial Backward Euler 25 20 15 10 0 2 4 6 Months of the Year 8 10 12 Figure 5 : A Comparative Chart Showing the Maximum Temperature Deduced by Various Schemes in one year Observing our choice Trapezoidal Scheme from Figure 5 above it is obviously showing that the temperature will be high from January till around May and it begin to decline from June till around October where it will rise slightly in November. Solution of Minimum Temperature Prediction Using Finite Difference Scheme Using equation (17) and sunshine hours value from Table 1 for the sixth month, we compute the predicted values for the different schemes. 𝑈 (𝑛+1) = 𝑅𝑒[𝜆𝑈 (𝑛) ] For Heun Scheme where 𝜆 = 0.5 + 𝑖 and 𝑈 (𝑛) = 23.2 for minimum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[23.2(0.5 + 𝑖)] = 𝑅𝑒[11.75 + 23.5𝑖] = 11.6 − 1 = 10.6 For Matsuno Scheme where 𝜆 = 𝑖 and 𝑈 (𝑛) = 23.2 for minimum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[23.2(𝑖)] = 𝑅𝑒[23.2𝑖] = 0 For Trapezoidal Scheme Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 927 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 where 𝜆 = 1 + 𝑖/1.25 and 𝑈 (𝑛) = 23.2 for minimum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[23.2(1 + 𝑖/1.25)] = 23.2 − 0.64 = 22.56 For Backward Scheme where 𝜆 = 0.5 + 0.125𝑖 and 𝑈 (𝑛) = 23.2 for minimum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[23.2(0.5 + 0.125𝑖)] = 11.75 − 0.015625 = 11.58 For Euler Scheme where 𝜆 = 1 + 𝑖 and 𝑈 (𝑛) = 23.2 for minimum temperature then 𝑈 (𝑛+1) = 𝑅𝑒[23.2(1 + 𝑖)] = 23.2 − 1 = 22.2 The results of predicted minimum temperature for all the 12 months of the year are shown in Table 7 Table 7: Minimum Temperature 2016 (TMin 2016) Months 1 2 3 4 5 6 7 8 9 10 11 12 𝜔 Wavelength 𝜆 Heun 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 0.5 +𝑖 Matsuno 𝑖 Trapezoidal 1 + 𝑖/1.25 𝑖 1 + 𝑖/1.25 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 1 + 𝑖/1.25 Backward 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 0.5 + 0.125𝑖 Amplitude 𝑈 (𝑛) TMin ‘15 19.3 Schemes 𝑈 (𝑛+1) (TMin 2016) Heun 8.65 Matsuno 0 Trapezoidal 18.66 Backward 9.6 Euler 18.3 1+𝑖 23.2 10.6 0 22.56 11.58 22.2 1+𝑖 25 11.5 0 24.36 12.48 24 1+𝑖 25.7 11.8 0 25.06 12.78 24.7 1+𝑖 24.6 11.3 0 23.96 12.3 23.6 1+𝑖 23.2 10.6 0 22.56 11.58 22.2 1+𝑖 22.3 10.2 0 21.66 11.2 21.3 1+𝑖 22.5 10.3 0 21.86 11.3 21.5 1+𝑖 22.2 10.1 0 21.56 11.1 21.2 1+𝑖 21.8 9.9 0 21.16 10.88 20.8 1+𝑖 21.6 9.8 0 20.96 10.78 20.6 1+𝑖 17.2 7.6 0 16.56 8.58 16.2 Euler 1+𝑖 From the above table, the selection of the scheme to represent the model forecasting for the Minimum Temperature for 2016 is based on the trend of the scheme whose result is closest to the previous year i.e. 2015 and hence among all five schemes in the table it is very obvious that aside Euler’s (Forward) Scheme which is the second most closest, the Trapezoidal Scheme is the most closest to the given Minimum Temperature in 2015. Hence we use the Minimum Temperature predicted using the Trapezoidal Scheme. Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com 928 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 A Comparative Chart Showing the Minimum Temperature Deduced by Various Scheme 26 Min Temp. Heun Trapezodial Backward Euler 24 Minimum Temperature for the Various Schemes 22 20 18 16 14 12 10 8 6 0 2 4 6 Months of the Year 8 10 12 Figure 6 : A Comparative Chart Showing the Minimum Temperature Deduced by Various Schemes in one year Observing our choice Trapezoidal Scheme from Figure 6 above it is obviously showing that the temperature will be fall from January till around May and it begin to rise from June till around October where it will fall slightly in November. Summary of Predicted Weather Data Set From Compatible Finite Difference Scheme Table 8: Compatible FDM Numerical Weather Prediction Year: 2016 Station: ABUJA, NG Elev: 343.1ft. Lat: 09.15oN Lon: 07.00oE STATIO N NUMBE R 65125 STATIO N NAME ELE V LAT LONG DAT E RelHu m TMA X TMI N RAINFA LL SUNSHI NE HRS Abuja 49.36 36.76 Abuja 61.36 37.06 65125 Abuja 20160 1 20160 2 20160 3 20160 4 61.36 35.96 18.6 6 22.5 6 24.3 6 25.0 6 -0.64 65125 09.24’ W 09.24’ W 09.24’ W 09.24’ W 34.86 Abuja 09.15’ N 09.15’ N 09.15’ N 09.15’ N 42.36 65125 343. 1 343. 1 343. 1 343. 1 Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com WIND DIRECTIO N 6.66 WIN D SPEE D 2.26 -0.04 6.86 3.06 N 6.86 7.56 2.86 NW 73.56 6.86 4.36 NE NE 929 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 65125 Abuja 65125 Abuja 65125 Abuja 65125 Abuja 65125 Abuja 65125 Abuja 65125 Abuja 65125 Abuja 343. 1 343. 1 343. 1 343. 1 343. 1 343. 1 343. 1 343. 1 09.15’ N 09.15’ N 09.15’ N 09.15’ N 09.15’ N 09.15’ N 09.15’ N 09.15’ N 09.24’ W 09.24’ W 09.24’ W 09.24’ W 09.24’ W 09.24’ W 09.24’ W 09.24’ W 20160 5 20160 6 20160 7 20160 8 20160 9 20161 0 20161 1 20161 2 75.36 35.16 80.36 29.57 85.36 28.06 86.36 28.06 82.36 28.86 77.36 29.36 63.36 33.06 35.36 34.36 23.9 6 22.5 6 21.6 6 21.8 6 21.5 6 21.1 6 20.9 6 16.5 6 108.56 6.76 4.26 NE 266.56 6.86 4.06 N 314.16 3.86 3.06 NE 277.66 4.56 3.56 NW 257.76 4.56 3.46 W 237.56 6.16 2.66 E Trace 8.56 2.36 W -0.64 8.16 2.56 NE Table 8 shows the values of the predicted weather data values obtained by using the trapezoidal scheme. This compared favourably with the real weather data values collected from Federal Airport Authority of Nigeria (FAAN) Abuja Station shown on Table 4.1. non-conservative shallow atmospheric weather equations. Intrenational Journal for Numerical III.CONCLUSION Methods in Fluids 63 (6), 701 - 724 Weather prediction for a particular station is mostly [2]. Cotter, C. and Ham, D., (2011). Numerical wave accurate in the advent of recursive use of previous propagation for the triangular PIDG-P2 finite predictions or measurement. This research has element pair. Journal of Computational Physics unveiled that studying the weather trends helps in 230 (8), 2806-2820. predicting future weather attenuation using numerical [3]. Cotter, C., Shipton, J., (2012). Mixed Finite solutions deduced by finite difference method. The finite difference method has been used to deduce Elements for Numerical Weather Prediction. Journal of Computational Physics 231 (21), compatible models for automated attenuation of 7076-7091. various parameters involved in the weather formation [4]. Cotter, C. J., Ham, D. A. and Pain, C.C., (2009) with the use of MATLAB in predicting future weather A trends. The derivation of the models based on the finite difference method gives a high level of element pair for shallow-water ocean modelling. Ocean Modelling 26, 86-90. significance. In conclusion, the weather prediction for [5]. mixed discontinuous/continuous finite Danilov, S., (2010). On Utility of Triangular C- a station (i.e. Abuja, Nigeria) was flexibly obtained grid accurately prior to the use of previous determined or Modeling of Large-Scale Ocean Ows. Ocean forecasted data and a compatible C-grid staggered Dynamics 60 (6), 1361-1369 finite difference method. [6]. Type Discretization for Numerical Durojaye M. O. and Odeyemi J.K (2019) Radial basis function – finite difference (RBF-FD) IV. REFERENCES approximations for two- dimensional heat equations. The Pacific Journal of Science and [1]. Comblen, R., Lambrechts, J., Remacle, J. F. and Legat, V., (2010). Practical evaluation of five partly discontinous finite element pairs for the Volume 6, Issue 3, May-June-2020 | http://ijsrcseit.com Technology 20(2), 43-49 [7]. L e Roux, D.Y. and Pouliout, B., (2008). Analysis of numerically induced oscillations in two- 930 Jacob Emmanuel et al Int J Sci Res CSE & IT, May-June-2020; 6 (3) : 915-931 dimensional finite-element shallow-water Cite this article as : models Part II: Free planetary waves. SIAM [8]. [9]. Journal on Scientific Computing 30 (4), 1971- Jacob Emmanuel, Ogunfiditimi F. O., Victor A. O, 1991. Odeyemi J.K, "Finite Difference Methods for Weather L e Roux, D.Y. and Sene A, Rostand V, and Prediction in Abuja, Nigeria", International Journal of Hanert E. (2005). 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