{"id":203,"date":"2019-01-10T23:26:04","date_gmt":"2019-01-11T07:26:04","guid":{"rendered":"http:\/\/gantovnik.com\/bio-tips\/?p=203"},"modified":"2019-01-10T23:49:17","modified_gmt":"2019-01-11T07:49:17","slug":"laplace-equation-2d","status":"publish","type":"post","link":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/laplace-equation-2d\/","title":{"rendered":"Laplace equation (2D)"},"content":{"rendered":"
import os\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.sparse as sp\nimport matplotlib as mpl\nimport scipy.sparse.linalg\nos.chdir(r'D:\\projects\\wordpress\\ex42')\nos.getcwd()\nN = 100\nu0_t, u0_b = 5, -5\nu0_l, u0_r = 3, -1\ndx = 1. \/ (N+1)\nA_1d = (sp.eye(N, k=-1) + sp.eye(N, k=1) - 4 * sp.eye(N))\/dx**2\nA = sp.kron(sp.eye(N), A_1d) + (sp.eye(N**2, k=-N) + sp.eye(N**2, k=N))\/dx**2\nprint(A)\nprint(A.nnz * 1.0\/ np.prod(A.shape) * 2000)\nd = np.zeros((N, N))\nd[0, :] += -u0_b \nd[-1, :] += -u0_t\nd[:, 0] += -u0_l\nd[:, -1] += -u0_r\nd = d.reshape(N**2) \/ dx**2\nu = sp.linalg.spsolve(A, d).reshape(N, N)\nU = np.vstack([np.ones((1, N+2)) * u0_b,\nnp.hstack([np.ones((N, 1)) * u0_l, u, np.ones((N, 1)) * u0_r]),\nnp.ones((1, N+2)) * u0_t])\nfig, ax = plt.subplots(1, 1, figsize=(8, 6))\nx = np.linspace(0, 1, N+2)\nX, Y = np.meshgrid(x, x)\nc = ax.pcolor(X, Y, U, vmin=-5, vmax=5, cmap=mpl.cm.get_cmap('RdBu_r'))\ncb = plt.colorbar(c, ax=ax)\nax.set_xlabel(r\"$x_1$\", fontsize=18)\nax.set_ylabel(r\"$x_2$\", fontsize=18)\ncb.set_label(r\"$u(x_1, x_2)$\", fontsize=18)\nfig.savefig(\"ch11-fdm-2d.pdf\")\nfig.tight_layout()\nplt.savefig(\"example42.png\", dpi=100)\nplt.show()\nplt.close()\n\n\n\"example42\"\n\nx = np.linspace(0, 1, N+2)\nX, Y = np.meshgrid(x, x)\nfig = plt.figure(figsize=(12, 5.5))\ncmap = mpl.cm.get_cmap('RdBu_r')\nax = fig.add_subplot(1, 2, 1)\np = ax.pcolor(X, Y, U, vmin=-5, vmax=5, cmap=cmap)\nax.set_xlabel(r\"$x_1$\", fontsize=18)\nax.set_ylabel(r\"$x_2$\", fontsize=18)\nax = fig.add_subplot(1, 2, 2, projection='3d')\np = ax.plot_surface(X, Y, U, vmin=-5, vmax=5, rstride=3, cstride=3, linewidth=0, cmap=cmap)\nax.set_xlabel(r\"$x_1$\", fontsize=16)\nax.set_ylabel(r\"$x_2$\", fontsize=16)\ncb = plt.colorbar(p, ax=ax, shrink=0.75)\ncb.set_label(r\"$u(x_1, x_2)$\", fontsize=18)\nplt.savefig(\"example42_2.png\", dpi=100)\nplt.show()\nplt.close()\n\n\n\"example42_2\"\n\n\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
import os import numpy as np import matplotlib.pyplot as plt import scipy.sparse as sp import matplotlib as mpl import scipy.sparse.linalg os.chdir(r’D:\\projects\\wordpress\\ex42′) os.getcwd() N = 100 u0_t, u0_b = 5, -5 u0_l, u0_r = 3, -1 dx = 1. \/ (N+1) A_1d = (sp.eye(N, k=-1) + sp.eye(N, k=1) – 4 * sp.eye(N))\/dx**2 A = sp.kron(sp.eye(N), A_1d) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","_lmt_disableupdate":"yes","_lmt_disable":"","jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-203","post","type-post","status-publish","format-standard","hentry","category-python"],"modified_by":null,"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p8bH0k-3h","jetpack_likes_enabled":true,"jetpack-related-posts":[{"id":200,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/heat-equation-1d\/","url_meta":{"origin":203,"position":0},"title":"Heat equation (1D)","author":"gantovnik","date":"2019-01-10","format":false,"excerpt":"import os import numpy as np import matplotlib.pyplot as plt os.chdir(r'D:\\projects\\wordpress\\ex41') os.getcwd() N = 5 u0 = 1 u1 = 2 dx = 1.0 \/ (N + 1) A = (np.eye(N, k=-1) - 2 * np.eye(N) + np.eye(N, k=1)) \/ dx**2 print(A) d = -5 * np.ones(N) d[0] -= u0\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example41","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example41.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example41.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example41.png?resize=525%2C300 1.5x"},"classes":[]},{"id":2166,"url":"https:\/\/gantovnik.com\/bio-tips\/2024\/05\/423-the-2d-diffusion-equation-applied-to-the-temperature-of-a-steel-circular-plate\/","url_meta":{"origin":203,"position":1},"title":"#423 The 2D diffusion equation applied to the temperature of a steel plate","author":"gantovnik","date":"2024-05-05","format":false,"excerpt":"","rel":"","context":"In "matplotlib"","block_context":{"text":"matplotlib","link":"https:\/\/gantovnik.com\/bio-tips\/category\/matplotlib\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/05\/ex423.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/05\/ex423.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/05\/ex423.png?resize=525%2C300&ssl=1 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to ODEs","author":"gantovnik","date":"2019-01-08","format":false,"excerpt":"\u00a0 import os import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl import sympy from IPython.display import display sympy.init_printing() mpl.rcParams['text.usetex'] = True import sympy os.chdir(r'D:\\projects\\wordpress\\ex33') os.getcwd() def plot_direction_field(x, y_x, f_xy, x_lim=(-5, 5), y_lim=(-5, 5), ax=None): f_np = sympy.lambdify((x, y_x), f_xy, 'numpy') x_vec = np.linspace(x_lim[0], x_lim[1], 20) y_vec\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example33","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example33.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example33.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example33.png?resize=525%2C300 1.5x"},"classes":[]},{"id":151,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/direction-fields\/","url_meta":{"origin":203,"position":5},"title":"Direction fields","author":"gantovnik","date":"2019-01-07","format":false,"excerpt":"","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example32","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example32.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example32.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example32.png?resize=525%2C300 1.5x"},"classes":[]}],"_links":{"self":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/203","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/comments?post=203"}],"version-history":[{"count":0,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/203\/revisions"}],"wp:attachment":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/media?parent=203"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/categories?post=203"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/tags?post=203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}