ó ·´Vc@sPddlZddlmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZdZdZeedJscCs"tjt|t|ƒdgƒS(sVA function which returns an np.array but less easy to read than f(z,t) below. i(RRR+tu_exact(tuR"((sw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pytf_localLscCst|t|ƒdgS(s'Simple to read function implementation i(R+R.(RR"((sw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pytfQsc Csçddlm}m}d}d}d}|d||dƒ}tjdƒ}tdƒ|dt|d ƒD],}'| |||"ƒ}(||#|(dd…d fƒ})||)d ƒ}*t|*ƒ}+t|"d |*||'||d |!dƒ|j| jd!t|!ƒƒtd"ƒ|'d krL|&j|,|+ƒn|+},|!d 9}!||||!d ƒ}"tj|"ƒ}#d }$x(|"D] }%||%ƒ|#|$<|$d }$qŽWq†W|&|| j<|d 7}q Wt|d#d$ƒtd%ƒtd&ƒt ƒ|!d |}!gtd |d ƒD]}'|!d |'^q}-tj!|-ƒ}-tƒx(|D] }.t|-tj!||.ƒƒqUWt"d'd(d d)|!d*dd+dƒt"d,d(d d)|!d*dd+dƒt"d-d(d d)|!d*dd+dƒt#|-d.d/ƒ|j$ƒ}|jd0ƒt|d#d$d1t%ƒt&d2ƒtd3ƒtd4ƒt'd t|-ƒd dgƒdS(5sv Test convergence rate of the methods, by using the Method of Manufactured solutions. The coefficient function f is chosen to be the normal distribution f = (1/(sigma*sqrt(2*pi)))*exp(-((t-mu)**2)/(2*sigma**2)). The ODE to be solved is than chosen to be: f''' + f''*f + f' = RHS, leading to to f''' = RHS - f''*f - f iÿÿÿÿ(R2RRBRFRGRH(RAtsymbolstdifftlambdify(tsqrttpis(solving equation f''' + f''*f + f' = RHSs$which lead to f''' = RHS - f''*f - fR"gà?gø¿g@iiicsItj|ƒ}|d|dˆ|ƒ|d|d|dg|(|S(Niii(RR(RMR"tyout(tRHS(sw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pyRÝs6gV瞯Ò f1 = C*h1^p - error1 = 0 error2 = C*h2^p --> f2 = C h2^p - error 2 = 0 where C is a constant h is the step length and p is the order, with use of a newton rhapson solver. In this case C and p are the unknowns, whereas h and error are knowns. The newton rhapson method is an iterative solver which take the form: xnew = xold - (J^-1)*F, where J is the Jacobi matrix and F is the residual funcion. x = [C, p]^T J = [[df1/dx1 df2/dx2], [df2/dx1 df2/dx2]] F = [f1, f2] This is very neatly done with use of the sympy module Args: error(list): list of calculated errors [error(h1), error(h2)] h(list): list of steplengths corresponding to the list of errors x0(list): list of starting (guessed) values for x Returns: x(array): iterated solution of x = [C, p] iÿÿÿÿ(tMatrixsC piþÿÿÿcsˆˆ|ˆ|ƒS(N((Rtj(tFRptx(sw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pyt jacobiElement©siiiii(RvRˆtinvRRRtdottround(terrorthtx0RˆtCtptf1tf2RŒtJacobit JacobiInvt JacobiInvfunctFfunctntJinvtxnew(RpRqRo(RŠR‹sw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pytNewton_solver_sympyˆs(    RR#R)(tlatexR‘t$R,sb--RJsg--RIsr--s${\epsilon}_{euler}$s${\epsilon}_{heun}$s${\epsilon}_{rk4}$s-log(h)s-log($\epsilon$)(gø¿g@('R3R2RBRvRARoRpRqRwRrRsR„RHRRxRR#R)RRtTrueR`R RR]R4RR^R5t set_titlet set_xlabelRt set_ylabelRRRŸRR R (6R2RBRARrRsR„RHR"RyRzR1R~RR€RR|R}RR:t schemes_errorR‘tNtdstfigtaxRLR;R9Rt legendListRRRRRhRžthtt eulerErrort heunErrortrk4ErrortC_eulertp_eulertC_heuntp_heuntC_rk4tp_rk4RŸt epsilon_eulertepsilon_euler_latext epsilon_heuntepsilon_heun_latext epsilon_rk4tepsilon_rk4_latexRlt LegendList((RuRpRqRosw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pytmanufactured_solution_Nonlinear-sª" 7 *'   (  !+5 .   c Csáddlm}tƒd}d}|d||dƒ}d}tttg}g}xM|D]E}|t||ƒ}t||dd…dfƒ|j|j ƒq[Wt|t ||ƒƒ|jd ƒt |d d d t ƒdS( s2Plot the solutions for the test schemes in schemesiÿÿÿÿ(R2gø?i2iig@Nt analyticalRSRTR\( R3R2RRR#R)R@RR]R5RERR`( R2R8R9RRR:RbR;R((sw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pytplot_ODEschemes_solutionsìs  ('R3Rtmatplotlib.pyplotRRRRRRRRRR R R R R RRRRtLNWDTtFNTRR#R)t__name__R+R,R.R0R1R>R@RERnRƒR»R½(((sw/Users/fredrikeikelandfossan/Documents/git/tkt4140_digital_compendium/digital_compendium/chapter1/src-ch1/ODEschemes.pyts. v          D o ¿