《FDTD electromagnetic field using MATLAB》讀書筆記 Figure 1.2
阿新 • • 發佈:2017-06-26
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函數f(x)用采樣間隔Δx=π/5進行采樣,使用向前差商、向後差商和中心差商三種公式來近似一階導數。
書中代碼:
%% ------------------------------------------------------------------------------ %% Output Info about this m-file fprintf(‘\n****************************************************************\n‘); fprintf(‘\n <FDTD 4 ElectroMagnetics with MATLAB Simulations> \n‘); fprintf(‘\n Figure 1.2 \n\n‘); time_stamp = datestr(now, 31); [wkd1, wkd2] = weekday(today, ‘long‘); fprintf(‘ Now is %20s, and it is %7s \n\n‘, time_stamp, wkd2); %% ------------------------------------------------------------------------------ % Create exact function and its derivative N_exact = 301; % number of sample points for exact function x_exact = linspace(0, 6*pi, N_exact); f_exact = sin(x_exact) .* exp(-0.3*x_exact); f_derivative_exact = cos(x_exact) .* exp(-0.3*x_exact) - 0.3*sin(x_exact).*exp(-0.3*x_exact); % plot exact function figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Figure 1.2.a‘); set(gcf,‘Color‘,‘white‘); plot(x_exact, f_exact, ‘k-‘, ‘linewidth‘, 1.5); set(gca, ‘fontsize‘, 12, ‘fontweight‘, ‘demi‘); axis([0 6*pi -1 1]); grid on; xlabel(‘$x$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); ylabel(‘$f(x)$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); title(‘Exact function‘); % create exact function for pi/5 sampleing peroid and % its finite difference derivatives N_a = 31; % number of points for pi/5 sampling period x_a = linspace(0, 6*pi, N_a); % [0, 6pi], row vector with 31 points f_a = sin(x_a) .* exp(-0.3*x_a); f_derivative_a = cos(x_a) .* exp(-0.3*x_a) - 0.3*sin(x_a) .* exp(-0.3*x_a); dx_a = pi/5; f_derivative_forward_a = zeros(1, N_a); % 1×31 zero matrix f_derivative_backward_a = zeros(1, N_a); f_derivative_central_a = zeros(1, N_a); f_derivative_forward_a(1:N_a-1) = (f_a(2:N_a)-f_a(1:N_a-1))/dx_a; f_derivative_backward_a(2:N_a) = (f_a(2:N_a)-f_a(1:N_a-1))/dx_a; f_derivative_central_a(2:N_a-1) = (f_a(3:N_a)-f_a(1:N_a-2))/(2*dx_a); % create exact function for pi/10 sampleing peroid and % its finite difference derivatives N_b = 61; % number of points for pi/10 sampling period x_b = linspace(0, 6*pi, N_b); f_b = sin(x_b) .* exp(-0.3*x_b); f_derivative_b = cos(x_b) .* exp(-0.3*x_b) - 0.3*sin(x_b) .* exp(-0.3*x_b); dx_b = pi/10; f_derivative_forward_b = zeros(1, N_b); f_derivative_backward_b = zeros(1, N_b); f_derivative_central_b = zeros(1, N_b); f_derivative_forward_b(1:N_b-1) = (f_b(2:N_b)-f_b(1:N_b-1))/dx_b; f_derivative_backward_b(2:N_b) = (f_b(2:N_b)-f_b(1:N_b-1))/dx_b; f_derivative_central_b(2:N_b-1) = (f_b(3:N_b)-f_b(1:N_b-2))/(2*dx_b); % plot exact derivative of the function and its finite difference % derivatives using pi/5 sampling period figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Figure 1.2.b‘); set(gcf,‘Color‘,‘white‘); plot(x_exact, f_derivative_exact, ‘k‘, ... x_a(1:N_a-1), f_derivative_forward_a(1:N_a-1), ‘b--‘, ... x_a(2:N_a), f_derivative_backward_a(2:N_a), ‘r-.‘, ... x_a(2:N_a-1), f_derivative_central_a(2:N_a-1), ‘:ms‘, ... ‘markersize‘, 4, ‘linewidth‘, 1.5); set(gca, ‘fontsize‘, 12, ‘fontweight‘, ‘demi‘); axis([0 6*pi -1 1]); grid on; legend(‘exact‘, ‘forward difference‘, ‘backward difference‘, ‘central difference‘); xlabel(‘$x$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); ylabel(‘$f‘‘(x)$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); text(pi, 0.6, ‘$\Delta x = \pi/5$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16, ‘backgroundcolor‘, ... ‘w‘, ‘edgecolor‘, ‘k‘); % plot error for finite difference derivatives % using pi/5 sampling period error_forward_a = f_derivative_a - f_derivative_forward_a; error_backward_a = f_derivative_a - f_derivative_backward_a; error_central_a = f_derivative_a - f_derivative_central_a; figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Figure 1.2.c‘); set(gcf,‘Color‘,‘white‘); plot(x_a(1:N_a-1), error_forward_a(1:N_a-1), ‘b--‘, ... x_a(2:N_a), error_backward_a(2:N_a), ‘r--‘, ... x_a(2:N_a-1), error_central_a(2:N_a-1), ‘:ms‘, ... ‘markersize‘, 4, ‘linewidth‘, 1.5); set(gca, ‘fontsize‘, 12, ‘fontweight‘, ‘demi‘); axis([0 6*pi -0.2 0.2]); grid on; legend(‘forward difference‘, ‘backward difference‘, ‘central difference‘); xlabel(‘$x$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); ylabel(‘error $[f‘‘(x)]$‘ , ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); text(pi, 0.15, ‘$\Delta x = \pi/5$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16, ... ‘backgroundcolor‘, ‘w‘, ‘edgecolor‘, ‘k‘); % plot error for finite difference derivatives % using pi/10 sampling period error_forward_b = f_derivative_b - f_derivative_forward_b; error_backward_b = f_derivative_b - f_derivative_backward_b; error_central_b = f_derivative_b - f_derivative_central_b; figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Figure 1.2.d‘); set(gcf,‘Color‘,‘white‘); plot(x_b(1:N_b-1), error_forward_b(1:N_b-1), ‘b--‘, ... x_b(2:N_b), error_backward_b(2:N_b), ‘r-.‘, ... x_b(2:N_b-1), error_central_b(2:N_b-1), ‘:ms‘, ... ‘markersize‘, 4, ‘linewidth‘, 1.5); set(gca, ‘fontsize‘, 12, ‘fontweight‘, ‘demi‘); axis([0 6*pi -0.2 0.2]); grid on; legend(‘forward difference‘, ‘backward difference‘, ‘central difference‘); xlabel(‘$x$‘, ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); ylabel(‘error $[f‘‘(x)]$‘ , ‘interpreter‘, ‘latex‘, ‘fontsize‘, 16); text(pi, 0.15, ‘$\Delta x = \pi/10$‘ , ‘interpreter‘, ... ‘latex‘, ‘fontsize‘, 16, ‘backgroundcolor‘, ‘w‘, ‘edgecolor‘, ‘k‘ );
運行結果:
上圖是函數圖形,看出振幅是指數衰減的。下圖是一階導數的精確值(公式計算)和三種差商近似結果。中心差商近似結果接近
精確值。
下圖是在Δx=π/5采樣間隔下,三種差商近似與精確值之間的誤差對比。可以看出中心差商近似的誤差最小。
下圖是Δx=π/10采樣間隔下,三種差商近似與精確值之間的誤差對比。可以看出中心差商近似的誤差最小。另外由於向前差商和
向後差商近似是1階精度,中心差商近似是2階精度,所以采樣間隔由π/5變成π/10後,向前差商和向後差商近似誤差變為原來的二分之一,
而中心差商近似誤差變為原來的四分之一。
《FDTD electromagnetic field using MATLAB》讀書筆記 Figure 1.2