def METHOD_NAME(self,beta1,beta2,index,Cdtype):
    start = timer()
    AD = self.get_left_matrix(beta1,beta2,self.pde.diffusion_coefficient,index,Cdtype)
    b = self.get_right_vector(beta1,index,self.pde.diffusion_coefficient,Cdtype)
    end = timer()
    print("Construct linear system time:", end - start)
    start = timer()
    self.uh[:] = spsolve(AD, b)
    end = timer()
    print("Solve time:", end-start)
    ls = {'AD':AD, 'b':b, 'solution':self.uh.copy()}
    return ls # return the linear system