Coverage for pySDC/projects/FastWaveSlowWave/plot_dispersion.py: 97%

1import matplotlib 

2 

3matplotlib.use('Agg') 

4 

5import matplotlib.pyplot as plt 

6import numpy as np 

7import sympy 

8from pylab import rcParams 

9 

10from pySDC.implementations.problem_classes.FastWaveSlowWave_0D import swfw_scalar 

11from pySDC.implementations.sweeper_classes.imex_1st_order import imex_1st_order 

12 

13from pySDC.implementations.problem_classes.acoustic_helpers.standard_integrators import dirk, rk_imex 

14 

15from pySDC.core.Step import step 

16 

17 

18def findomega(stab_fh): 

19 assert np.array_equal(np.shape(stab_fh), [2, 2]), 'Not 2x2 matrix...' 

20 omega = sympy.Symbol('omega') 

21 func = (sympy.exp(-1j * omega) - stab_fh[0, 0]) * (sympy.exp(-1j * omega) - stab_fh[1, 1]) - stab_fh[ 

22 0, 1 

23 ] * stab_fh[1, 0] 

24 solsym = sympy.solve(func, omega) 

25 sol0 = complex(solsym[0]) 

26 sol1 = complex(solsym[1]) 

27 if sol0.real >= 0: 

28 sol = sol0 

29 elif sol1.real >= 0: 

30 sol = sol1 

31 else: 

32 print("Two roots with real part of same sign...") 

33 sol = sol0 

34 return sol 

35 

36 

37def compute_and_plot_dispersion(Nsamples=15, K=3): 

38 """ 

39 Function to compute and plot the dispersion relation 

40 

41 Args: 

42 Nsamples: number of samples for testing 

43 K: number of iterations as well as order 

44 """ 

45 problem_params = dict() 

46 # SET VALUE FOR lambda_slow AND VALUES FOR lambda_fast ### 

47 problem_params['lambda_s'] = np.array([0.0]) 

48 problem_params['lambda_f'] = np.array([0.0]) 

49 problem_params['u0'] = 1.0 

50 

51 # initialize sweeper parameters 

52 sweeper_params = dict() 

53 # SET TYPE AND NUMBER OF QUADRATURE NODES ### 

54 sweeper_params['quad_type'] = 'RADAU-RIGHT' 

55 sweeper_params['do_coll_update'] = True 

56 sweeper_params['num_nodes'] = 3 

57 

58 # initialize level parameters 

59 level_params = dict() 

60 level_params['dt'] = 1.0 

61 

62 # fill description dictionary for easy step instantiation 

63 description = dict() 

64 description['problem_class'] = swfw_scalar # pass problem class 

65 description['problem_params'] = problem_params # pass problem parameters 

66 description['sweeper_class'] = imex_1st_order # pass sweeper 

67 description['sweeper_params'] = sweeper_params # pass sweeper parameters 

68 description['level_params'] = level_params # pass level parameters 

69 description['step_params'] = dict() # pass step parameters 

70 

71 # ORDER OF DIRK/IMEX IS EQUAL TO NUMBER OF ITERATIONS AND THUS ORDER OF SDC ### 

72 dirk_order = K 

73 

74 c_speed = 1.0 

75 U_speed = 0.05 

76 

77 # now the description contains more or less everything we need to create a step 

78 S = step(description=description) 

79 

80 L = S.levels[0] 

81 

82 # u0 = S.levels[0].prob.u_exact(t0) 

83 # S.init_step(u0) 

84 QE = L.sweep.QE[1:, 1:] 

85 QI = L.sweep.QI[1:, 1:] 

86 Q = L.sweep.coll.Qmat[1:, 1:] 

87 nnodes = L.sweep.coll.num_nodes 

88 dt = L.params.dt 

89 

90 k_vec = np.linspace(0, np.pi, Nsamples + 1, endpoint=False) 

91 k_vec = k_vec[1:] 

92 phase = np.zeros((3, Nsamples)) 

93 amp_factor = np.zeros((3, Nsamples)) 

94 

95 for i in range(0, np.size(k_vec)): 

96 Cs = -1j * k_vec[i] * np.array([[0.0, c_speed], [c_speed, 0.0]], dtype='complex') 

97 Uadv = -1j * k_vec[i] * np.array([[U_speed, 0.0], [0.0, U_speed]], dtype='complex') 

98 

99 LHS = np.eye(2 * nnodes) - dt * (np.kron(QI, Cs) + np.kron(QE, Uadv)) 

100 RHS = dt * (np.kron(Q, Uadv + Cs) - np.kron(QI, Cs) - np.kron(QE, Uadv)) 

101 

102 LHSinv = np.linalg.inv(LHS) 

103 Mat_sweep = np.linalg.matrix_power(LHSinv.dot(RHS), K) 

104 for k in range(0, K): 

105 Mat_sweep = Mat_sweep + np.linalg.matrix_power(LHSinv.dot(RHS), k).dot(LHSinv) 

106 ## 

107 # ---> The update formula for this case need verification!! 

108 update = dt * np.kron(L.sweep.coll.weights, Uadv + Cs) 

109 

110 y1 = np.array([1, 0], dtype='complex') 

111 y2 = np.array([0, 1], dtype='complex') 

112 e1 = np.kron(np.ones(nnodes), y1) 

113 stab_fh_1 = y1 + update.dot(Mat_sweep.dot(e1)) 

114 e2 = np.kron(np.ones(nnodes), y2) 

115 stab_fh_2 = y2 + update.dot(Mat_sweep.dot(e2)) 

116 stab_sdc = np.column_stack((stab_fh_1, stab_fh_2)) 

117 

118 # Stability function of backward Euler is 1/(1-z); system is y' = (Cs+Uadv)*y 

119 # stab_ie = np.linalg.inv( np.eye(2) - step.status.dt*(Cs+Uadv) ) 

120 

121 # For testing, insert exact stability function exp(-dt*i*k*(Cs+Uadv) 

122 # stab_fh = la.expm(Cs+Uadv) 

123 

124 dirkts = dirk(Cs + Uadv, dirk_order) 

125 stab_fh1 = dirkts.timestep(y1, 1.0) 

126 stab_fh2 = dirkts.timestep(y2, 1.0) 

127 stab_dirk = np.column_stack((stab_fh1, stab_fh2)) 

128 

129 rkimex = rk_imex(M_fast=Cs, M_slow=Uadv, order=K) 

130 stab_fh1 = rkimex.timestep(y1, 1.0) 

131 stab_fh2 = rkimex.timestep(y2, 1.0) 

132 stab_rk_imex = np.column_stack((stab_fh1, stab_fh2)) 

133 

134 sol_sdc = findomega(stab_sdc) 

135 sol_dirk = findomega(stab_dirk) 

136 sol_rk_imex = findomega(stab_rk_imex) 

137 

138 # Now solve for discrete phase 

139 phase[0, i] = sol_sdc.real / k_vec[i] 

140 amp_factor[0, i] = np.exp(sol_sdc.imag) 

141 phase[1, i] = sol_dirk.real / k_vec[i] 

142 amp_factor[1, i] = np.exp(sol_dirk.imag) 

143 phase[2, i] = sol_rk_imex.real / k_vec[i] 

144 amp_factor[2, i] = np.exp(sol_rk_imex.imag) 

145 

146 rcParams['figure.figsize'] = 1.5, 1.5 

147 fs = 8 

148 fig = plt.figure() 

149 plt.plot(k_vec, (U_speed + c_speed) + np.zeros(np.size(k_vec)), '--', color='k', linewidth=1.5, label='Exact') 

150 plt.plot(k_vec, phase[1, :], '-', color='g', linewidth=1.5, label='DIRK(' + str(dirkts.order) + ')') 

151 plt.plot( 

152 k_vec, 

153 phase[2, :], 

154 '-+', 

155 color='r', 

156 linewidth=1.5, 

157 label='IMEX(' + str(rkimex.order) + ')', 

158 markevery=(2, 3), 

159 mew=1.0, 

160 ) 

161 plt.plot( 

162 k_vec, 

163 phase[0, :], 

164 '-o', 

165 color='b', 

166 linewidth=1.5, 

167 label='SDC(' + str(K) + ')', 

168 markevery=(1, 3), 

169 markersize=fs / 2, 

170 ) 

171 plt.xlabel('Wave number', fontsize=fs, labelpad=0.25) 

172 plt.ylabel('Phase speed', fontsize=fs, labelpad=0.5) 

173 plt.xlim([k_vec[0], k_vec[-1:]]) 

174 plt.ylim([0.0, 1.1 * (U_speed + c_speed)]) 

175 fig.gca().tick_params(axis='both', labelsize=fs) 

176 plt.legend(loc='lower left', fontsize=fs, prop={'size': fs - 2}) 

177 plt.xticks([0, 1, 2, 3], fontsize=fs) 

178 filename = 'data/phase-K' + str(K) + '-M' + str(sweeper_params['num_nodes']) + '.png' 

179 plt.gcf().savefig(filename, bbox_inches='tight') 

180 

181 fig = plt.figure() 

182 plt.plot(k_vec, 1.0 + np.zeros(np.size(k_vec)), '--', color='k', linewidth=1.5, label='Exact') 

183 plt.plot(k_vec, amp_factor[1, :], '-', color='g', linewidth=1.5, label='DIRK(' + str(dirkts.order) + ')') 

184 plt.plot( 

185 k_vec, 

186 amp_factor[2, :], 

187 '-+', 

188 color='r', 

189 linewidth=1.5, 

190 label='IMEX(' + str(rkimex.order) + ')', 

191 markevery=(2, 3), 

192 mew=1.0, 

193 ) 

194 plt.plot( 

195 k_vec, 

196 amp_factor[0, :], 

197 '-o', 

198 color='b', 

199 linewidth=1.5, 

200 label='SDC(' + str(K) + ')', 

201 markevery=(1, 3), 

202 markersize=fs / 2, 

203 ) 

204 plt.xlabel('Wave number', fontsize=fs, labelpad=0.25) 

205 plt.ylabel('Amplification factor', fontsize=fs, labelpad=0.5) 

206 fig.gca().tick_params(axis='both', labelsize=fs) 

207 plt.xlim([k_vec[0], k_vec[-1:]]) 

208 plt.ylim([k_vec[0], k_vec[-1:]]) 

209 plt.legend(loc='lower left', fontsize=fs, prop={'size': fs - 2}) 

210 plt.gca().set_ylim([0.0, 1.1]) 

211 plt.xticks([0, 1, 2, 3], fontsize=fs) 

212 filename = 'data/ampfactor-K' + str(K) + '-M' + str(sweeper_params['num_nodes']) + '.png' 

213 plt.gcf().savefig(filename, bbox_inches='tight') 

214 

215 

216if __name__ == "__main__": 

217 compute_and_plot_dispersion()