Burgers_PseudoSpectral_Solver.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "f494a734",
   "metadata": {},
   "outputs": [],
   "source": [
    "'''Solving Burgers eqn ut+u*ux=nu*uxx'''\n",
    "'''I will start with dealiased initial condition and integrate in fourier space to reduce computational cost'''\n",
    "\n",
    "import numpy as np\n",
    "from numpy.fft import fft,ifft,fftfreq\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import integrate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "c7122167",
   "metadata": {},
   "outputs": [],
   "source": [
    "#RK-4 method function with parameters iterate of u at present time-upand a get-rhs function f and time  step dt\n",
    "\n",
    "def rk4(up,f,dt):\n",
    "    #co-efficients of RK-4 time step\n",
    "    k1=f(up) \n",
    "    k2=f(up+dt*(k1/2))\n",
    "    k3=f(up+dt*(k2/2))\n",
    "    k4=f(up+dt*k3)\n",
    "    un=up+(1/6)*dt*(k1+2*k2+2*k3+k4)\n",
    "    return un\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "fec575c6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.5968789280360625e-08\n"
     ]
    }
   ],
   "source": [
    "#checking my rk-4 on a simple ode du/dt=u^(1/3),u(0)=1\n",
    "\n",
    "dt=0.1 \n",
    "t=np.arange(0,1,dt) #time points\n",
    "N=len(t) #numper of time points\n",
    "actsol=((2*t+3)/3)**(3/2) #actual solution\n",
    "\n",
    "#applying our created rk-4\n",
    "\n",
    "compsol=np.zeros_like(t)\n",
    "compsol[0]=1\n",
    "for i in range(1,N):\n",
    "    compsol[i]=rk4(compsol[i-1],lambda x: x**(1/3),dt)\n",
    "\n",
    "#checking error\n",
    "\n",
    "error=np.max(np.abs(actsol-compsol))\n",
    "print(error)\n",
    "\n",
    "#Ya satisfies at least O(dt**4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "f6f6e731",
   "metadata": {},
   "outputs": [],
   "source": [
    "#writing a get-rhs fn of burgers with dealias filter\n",
    "#parameters -function u,viscosity nu,wavenumber array k,dealias filer dealias\n",
    "\n",
    "def getrhs(u,nu,k,dealias):\n",
    "    uk=fft(u) #fft of u\n",
    "    uxk=1j*k*uk #fft of ux\n",
    "    ux=np.real(ifft(uxk)) #first space derivative of u\n",
    "    uxxk=-(k**2)*uk #discrete fourier transform of uxx\n",
    "    uxx=np.real(ifft(uxxk))\n",
    "    nlp=u*ux #non-linear term with dealiased u- a blow is happening here with large N,why ?\n",
    "    nlpk=fft(nlp) #fft of non-linear term\n",
    "    nlpkde=dealias*nlpk #dealiased fft of nlp\n",
    "    nlpde=np.real(ifft(nlpkde))\n",
    "    return -nlpde+nu*uxx\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "a58f06bb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.070255078479022e-11\n"
     ]
    }
   ],
   "source": [
    "#checking if the function works\n",
    "\n",
    "N=2**15\n",
    "dx=2*np.pi/N\n",
    "x=np.arange(0,2*np.pi,dx)\n",
    "k=fftfreq(N,1/N)\n",
    "dealias=np.abs(k)<N/3\n",
    "u=1+np.cos(x)+np.cos(2*x)\n",
    "ux=-np.sin(x)-2*np.sin(2*x)\n",
    "uxx=-np.cos(x)-4*np.cos(2*x)\n",
    "nu=7*(10**(-6))\n",
    "computedrhs=getrhs(u,nu,k,dealias)\n",
    "actualrhs=-u*ux+nu*uxx\n",
    "error=np.max(np.abs(computedrhs-actualrhs))\n",
    "print(error)\n",
    "\n",
    "#Yo, not better error :( \n",
    "\n",
    "#correct one"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "65c1e6a1",
   "metadata": {},
   "outputs": [],
   "source": [
    "#total energy,parameters-u,L,N\n",
    "\n",
    "def energy(u,L,N):\n",
    "    dx=L/N #space spacing\n",
    "    x=np.arange(0,L,dx) #space points\n",
    "    E=np.abs(u)**2 #velocity squared array\n",
    "    return integrate.simpson(E,x) #total energy using simpson integral\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7dd9b3fe",
   "metadata": {},
   "outputs": [],
   "source": [
    "#energy spectrum\n",
    "\n",
    "def energyspectrum(u,k):\n",
    "    uk=fft(u) #fourier transform of u\n",
    "    Ek=np.abs(uk)**2 #energy spectrum\n",
    "    kmax=int(np.max(k)) #max positive-k\n",
    "    kp=k[0:kmax+1] #positive part of k\n",
    "    Ek=Ek[0:kmax+1] #Ek for only positive k values\n",
    "    return kp,Ek\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cdeccba5",
   "metadata": {},
   "outputs": [],
   "source": [
    "# transfer function\n",
    "\n",
    "def transferfunction(u,k,dealias):\n",
    "    uk=fft(u) #fft of u\n",
    "    uxk=1j*k*uk #fft of ux\n",
    "    ux=np.real(ifft(uxk)) #first space derivative of u\n",
    "    nlp=u*ux #non-linear term with dealiased u- a blow is happening here with large N,why ?\n",
    "    nlpk=fft(nlp) #fft of non-linear term\n",
    "    nlpkde=dealias*nlpk #dealiased fft of nlp\n",
    "    Tk=-nlpkde*np.conj(uk)-np.conj(nlpkde)*uk #the transfer function\n",
    "    Tk=np.real(Tk)\n",
    "    kmax=int(np.max(k)) #max in k\n",
    "    kp=k[0:kmax+1] #positive part of k\n",
    "    Tkp=Tk[0:kmax+1] #positive part of Tk\n",
    "    return kp,Tkp\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "60b253d5",
   "metadata": {},
   "outputs": [],
   "source": [
    "#total flux\n",
    "\n",
    "def flux(u,k,dealias):\n",
    "    kp,Tkp=transferfunction(u,k,dealias)\n",
    "    kmax=int(np.max(k))\n",
    "    pikp=np.zeros_like(kp)\n",
    "    for i in range(kmax):\n",
    "        pikp[i]=np.sum(Tkp[i:kmax+1])\n",
    "    return kp,pikp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "aef980dc",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "'''solving burgers with periodic boundary conditions using pseudospectral method without dealias and \n",
    "plotting total energy and energy spectrum'''\n",
    "\n",
    "#parameters - interval:[0,L],resolution:N,viscous parameter:nu,maximum time:tmax,timestep:dt,time save point step: s,solution number to be saved:num\n",
    "\n",
    "def burger_solve(L,N,nu,u0,Tmax,dt,s,no):\n",
    "    \n",
    "    #initialization\n",
    "    \n",
    "    dx=L/N #space-spacing\n",
    "    x=np.arange(0,L,dx) #space points\n",
    "    k=fftfreq(N,1/N)*(2*np.pi)/L #wave numbers\n",
    "    \n",
    "    kmax=int(np.max(k)) #max positive-k\n",
    "    kp=k[0:kmax+1] #positive part of k\n",
    "\n",
    "    #v=np.vectorize(u0) #vectorizing initial boundary function\n",
    "    #uin=v(x) #initial function array\n",
    "    dealias=np.abs(k)<N/3\n",
    "    \n",
    "    Nt=round(Tmax/dt) #number of time steps\n",
    "    t=np.arange(0,Tmax,dt) #time points\n",
    "    \n",
    "    #the rk-4 rhs function\n",
    "    \n",
    "    def f(u):\n",
    "        return getrhs(u,nu,k,dealias)\n",
    "    \n",
    "    #the rk-4 time stepping\n",
    "        \n",
    "    u=u0 #solution array\n",
    "    \n",
    "    np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/solution/u'+str(no)+'_0.npy',u)\n",
    "    \n",
    "    #energy time series\n",
    "    \n",
    "    E=np.zeros_like(t)\n",
    "    \n",
    "    E[0]=energy(u,L,N) #initial energy\n",
    "    \n",
    "    #k,energy spectrum\n",
    "    \n",
    "    Ek=energyspectrum(u,k)\n",
    "    \n",
    "    np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energyspectrum/Ek'+str(no)+'_0.npy',Ek)\n",
    "    \n",
    "    #transfer function\n",
    "    \n",
    "    Tk=transferfunction(u,k,dealias)\n",
    "    \n",
    "    np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/transferfunction/Tk'+str(no)+'_0.npy',Tk)\n",
    "    \n",
    "    #derivative of energy spectrum for checking that my Tk term is correct\n",
    "    \n",
    "    dEkp=np.zeros_like(kp) #initializing derivative \n",
    "    \n",
    "    #k,flux\n",
    "    \n",
    "    pik=flux(u,k,dealias)\n",
    "    \n",
    "    np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/flux/pik'+str(no)+'_0.npy',pik)\n",
    "    \n",
    "    for i in range(1,Nt):\n",
    "        \n",
    "        print(str(i)+\" \",end='') #checking that the code is running\n",
    "        \n",
    "        u=rk4(u,f,dt) #updated solution\n",
    "        \n",
    "        E[i]=energy(u,L,N) #energy at ith timestep\n",
    "        \n",
    "        Ekn=energyspectrum(u,k) #new k,energy spectrum\n",
    "        \n",
    "        dEkp=(Ekn[1]-Ek[1])/dt #derivative at time (i-1)\n",
    "        \n",
    "        Ek=Ekn #updating energy spectrum after computing derivative\n",
    "        \n",
    "        Tk=transferfunction(u,k,dealias) # updated k,transferfunction\n",
    "        \n",
    "        pik=flux(u,k,dealias) # updated k,flux\n",
    "        \n",
    "        #saving u,Ek,pik\n",
    "        \n",
    "        if i%s==0:\n",
    "            \n",
    "            np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/solution/u'+str(no)+'_'+str(i)+'.npy',u)\n",
    "            np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energyspectrum/Ek'+str(no)+'_'+str(i)+'.npy',Ek)\n",
    "            np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/transferfunction/Tk'+str(no)+'_'+str(i)+'.npy',Tk)\n",
    "            np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/flux/pik'+str(no)+'_'+str(i)+'.npy',pik)\n",
    "        \n",
    "        #saving energy spectrum derivative\n",
    "            \n",
    "        if int(i-1)%s==0:\n",
    "            np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energyspectrumderivative/dEk'+str(no)+'_'+str(int(i-1))+'.npy',(kp,dEkp))\n",
    "            \n",
    "            \n",
    "    np.save('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energy/E'+str(no)+'.npy',E)\n",
    "           \n",
    "            "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2662bdbd",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "#plotting solution\n",
    "\n",
    "#parameters -L,N,dt,Tmax,s\n",
    "\n",
    "def solutionplot(L,N,dt,Tmax,s,no,nu):\n",
    "    \n",
    "    dx=L/N #space-spacing\n",
    "    x=np.arange(0,L,dx) #space points\n",
    "    k=fftfreq(N,1/N)*(2*np.pi)/L #wave numbers\n",
    "    \n",
    "    Nt=round(Tmax/dt) #number of time steps\n",
    "    t=np.arange(0,Tmax,dt) #time points\n",
    "    \n",
    "    for i in range(0,Nt):\n",
    "        \n",
    "        print(str(i)+\" \",end='')\n",
    "        \n",
    "        if i%s==0:\n",
    "            \n",
    "            ti=t[i] #present time\n",
    "            u=np.load('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/solution/u'+str(no)+'_'+str(i)+'.npy')\n",
    "            \n",
    "            plt.plot(x,u)\n",
    "            plt.title('at time '+str(ti)+'nu='+str(nu)+'N='+str(N)+'dt='+str(dt))\n",
    "            plt.xlabel('x-axis')\n",
    "            plt.ylabel('solution u(x)')\n",
    "            plt.savefig('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/solutionplot/u'+str(no)+'_'+str(i)+'.png',dpi=300)\n",
    "            plt.close()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "911baa13",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting total energy \n",
    "\n",
    "def energyplot(L,N,dt,Tmax,s,no,nu):\n",
    "    \n",
    "    Nt=round(Tmax/dt) #number of time steps\n",
    "    t=np.arange(0,Tmax,dt) #time points\n",
    "    \n",
    "    E=np.load('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energy/E'+str(no)+'.npy')\n",
    "    \n",
    "    plt.plot(t,E)\n",
    "    plt.title('nu='+str(nu)+'N='+str(N)+'dt='+str(dt))\n",
    "    plt.xlabel('time t')\n",
    "    plt.ylabel('Total energy')\n",
    "    plt.savefig('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energyplot/E'+str(no)+'.png',dpi=300)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "24530820",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting energyspectrum\n",
    "\n",
    "#parameters -L,N,dt,Tmax,s\n",
    "\n",
    "def energyspectrumplot(L,N,dt,Tmax,s,no,nu):\n",
    "    \n",
    "    dx=L/N #space-spacing\n",
    "    x=np.arange(0,L,dx) #space points\n",
    "    k=fftfreq(N,1/N)*(2*np.pi)/L #wave numbers\n",
    "    \n",
    "    Nt=round(Tmax/dt) #number of time steps\n",
    "    t=np.arange(0,Tmax,dt) #time points\n",
    "    \n",
    "    for i in range(0,Nt):\n",
    "        \n",
    "        print(str(i)+\" \",end='') #checking run of code\n",
    "        \n",
    "        if i%s==0:\n",
    "            \n",
    "            ti=t[i] #present time\n",
    "            kp,Ekp=np.load('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energyspectrum/Ek'+str(no)+'_'+str(i)+'.npy')\n",
    "            \n",
    "            plt.loglog(kp,Ekp,'r')\n",
    "            plt.loglog(kp,kp**(-2),'b')\n",
    "            plt.title('at time '+str(ti)+'nu='+str(nu)+'N='+str(N)+'dt='+str(dt))\n",
    "            plt.xlabel('logk')\n",
    "            plt.ylabel('logEk')\n",
    "            #plt.ylim([10**(-7),None])\n",
    "            plt.savefig('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energyspectrumplot/Ek'+str(no)+'_'+str(i)+'.png',dpi=300)\n",
    "            plt.close()\n",
    "            \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "662101f4",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting transfer function and comparing with energyspectrumderivaytive to check accuracy\n",
    "\n",
    "def transferfunctionplot(L,N,dt,Tmax,s,no,nu):\n",
    "    \n",
    "    dx=L/N #space-spacing\n",
    "    x=np.arange(0,L,dx) #space points\n",
    "    k=fftfreq(N,1/N)*(2*np.pi)/L #wave numbers\n",
    "    \n",
    "    Nt=round(Tmax/dt) #number of time steps\n",
    "    t=np.arange(0,Tmax,dt) #time points\n",
    "    \n",
    "    for i in range(0,Nt):\n",
    "        \n",
    "        print(str(i)+\" \",end='') #checking run of code\n",
    "        \n",
    "        if i%s==0:\n",
    "            \n",
    "            ti=t[i] #present time\n",
    "            kp,Tkp=np.load('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/transferfunction/Tk'+str(no)+'_'+str(i)+'.npy')\n",
    "            kp,dEkp=np.load('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/energyspectrumderivative/dEk'+str(no)+'_'+str(i)+'.npy')\n",
    "            \n",
    "            plt.semilogx(kp,Tkp,label='transfer function')\n",
    "            plt.semilogx(kp,dEkp,label='energy spectrum derivative')\n",
    "            plt.legend()\n",
    "            plt.title('at time '+str(ti)+'nu='+str(nu)+'N='+str(N)+'dt='+str(dt))\n",
    "            plt.xlabel('logk')\n",
    "            plt.ylabel('Tk')\n",
    "            plt.savefig('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/transferfunctionvsenergyspectrumderivativeplot/dEk'+str(no)+'Tk'+str(no)+'_'+str(i)+'.png',dpi=300)\n",
    "            plt.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "341dff42",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting flux\n",
    "\n",
    "def fluxplot(L,N,dt,Tmax,s,no,nu):\n",
    "    \n",
    "    dx=L/N #space-spacing\n",
    "    x=np.arange(0,L,dx) #space points\n",
    "    k=fftfreq(N,1/N)*(2*np.pi)/L #wave numbers\n",
    "    \n",
    "    Nt=round(Tmax/dt) #number of time steps\n",
    "    t=np.arange(0,Tmax,dt) #time points\n",
    "    \n",
    "    for i in range(0,Nt):\n",
    "        \n",
    "        print(str(i)+\" \",end='') #checking run of code\n",
    "        \n",
    "        if i%s==0:\n",
    "            \n",
    "            ti=t[i] #present time\n",
    "            kp,pikp=np.load('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/flux/pik'+str(no)+'_'+str(i)+'.npy')\n",
    "            \n",
    "            plt.semilogx(kp,pikp)\n",
    "            plt.title('at time '+str(ti)+'nu='+str(nu)+'N='+str(N)+'dt='+str(dt))\n",
    "            plt.xlabel('logk')\n",
    "            plt.ylabel('pik')\n",
    "            plt.savefig('/localhome/souryajit/Desktop/mycodes/fftcodes/burgersrandomplots/fluxplot/pik'+str(no)+'_'+str(i)+'.png',dpi=300)\n",
    "            plt.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f276d235",
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "#running the solver for inviscid/viscous\n",
    "\n",
    "L=2*np.pi\n",
    "N=2**9\n",
    "nu=10**(-2)\n",
    "dx=L/N\n",
    "x=np.arange(0,L,dx)\n",
    "u0=1+np.cos(x)+np.cos(2*x)\n",
    "\n",
    "Tmax=2\n",
    "dt=10**(-3)\n",
    "s=10**2\n",
    "\n",
    "\n",
    "burger_solve(L,N,nu,u0,Tmax,dt,s)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "551ec8f4",
   "metadata": {},
   "outputs": [],
   "source": [
    "#running solver for random initialization for first 6 wave numbers\n",
    "\n",
    "L=2*np.pi\n",
    "\n",
    "N=2**15\n",
    "\n",
    "nu=4*10**(-4)\n",
    "\n",
    "k=fftfreq(N,1./N)\n",
    "\n",
    "#fft of inital data\n",
    "\n",
    "u0k=np.zeros_like(k,dtype='complex_')\n",
    "\n",
    "rand_array_real=np.random.rand(6)\n",
    "\n",
    "rand_array_imaginary=np.random.rand(6)\n",
    "\n",
    "u0k[0]=rand_array_real[0]+1j*rand_array_imaginary[0]\n",
    "\n",
    "for i in range(1,6):\n",
    "    u0k[i]=rand_array_real[i]+1j*rand_array_imaginary[i]\n",
    "    u0k[N-i]=rand_array_real[i]-1j*rand_array_imaginary[i]\n",
    "    \n",
    "u0k=N*u0k\n",
    "    \n",
    "#initial solution\n",
    "\n",
    "u0=np.real(ifft(u0k))\n",
    "\n",
    "Tmax=2\n",
    "\n",
    "dt=10**(-5)\n",
    "\n",
    "s=10**4\n",
    "\n",
    "no=18\n",
    "\n",
    "burger_solve(L,N,nu,u0,Tmax,dt,s,no)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cd2c780b",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting solution for inviscid/viscous\n",
    "\n",
    "L=2*np.pi\n",
    "N=2**15\n",
    "Tmax=2\n",
    "dt=10**(-5)\n",
    "s=10**4\n",
    "nu=4*10**(-4)\n",
    "no=18\n",
    "\n",
    "solutionplot(L,N,dt,Tmax,s,no,nu)\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "746e0fb0",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting energy time series for inviscid/viscous\n",
    "\n",
    "L=2*np.pi\n",
    "N=2**15\n",
    "Tmax=2\n",
    "dt=10**(-5)\n",
    "s=10**4\n",
    "nu=4*10**(-4)\n",
    "no=18\n",
    "energyplot(L,N,dt,Tmax,s,no,nu)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "aee3d7d7",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting energyspectrum for inviscid/viscous\n",
    "\n",
    "L=2*np.pi\n",
    "N=2**15\n",
    "Tmax=2\n",
    "dt=10**(-5)\n",
    "s=10**4\n",
    "nu=4*10**(-4)\n",
    "no=18\n",
    "\n",
    "energyspectrumplot(L,N,dt,Tmax,s,no,nu)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4896175b",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting transfer function vs energy spectrum derivative for inviscid case to check accuracy\n",
    "\n",
    "L=2*np.pi\n",
    "N=2**15\n",
    "Tmax=2\n",
    "dt=10**(-5)\n",
    "s=10**4\n",
    "nu=4*10**(-4)\n",
    "no=18\n",
    "\n",
    "\n",
    "transferfunctionplot(L,N,dt,Tmax,s,no,nu)\n",
    "\n",
    "'''initially there is an error in the small w nos for time t=0 which I think because \n",
    "we multiplied a dealias to the non linear term'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f8ce6720",
   "metadata": {},
   "outputs": [],
   "source": [
    "#plotting flux for inviscid/viscous\n",
    "\n",
    "L=2*np.pi\n",
    "N=2**15\n",
    "Tmax=2\n",
    "dt=10**(-5)\n",
    "s=10**4\n",
    "nu=4*10**(-4)\n",
    "no=18\n",
    "\n",
    "\n",
    "fluxplot(L,N,dt,Tmax,s,no,nu)\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d4e39c21",
   "metadata": {},
   "outputs": [],
   "source": [
    "%timeit burger_solve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "48a5043f",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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