462 research outputs found
Oscillation-free method for semilinear diffusion equations under noisy initial conditions
Noise in initial conditions from measurement errors can create unwanted
oscillations which propagate in numerical solutions. We present a technique of
prohibiting such oscillation errors when solving initial-boundary-value
problems of semilinear diffusion equations. Symmetric Strang splitting is
applied to the equation for solving the linear diffusion and nonlinear
remainder separately. An oscillation-free scheme is developed for overcoming
any oscillatory behavior when numerically solving the linear diffusion portion.
To demonstrate the ills of stable oscillations, we compare our method using a
weighted implicit Euler scheme to the Crank-Nicolson method. The
oscillation-free feature and stability of our method are analyzed through a
local linearization. The accuracy of our oscillation-free method is proved and
its usefulness is further verified through solving a Fisher-type equation where
oscillation-free solutions are successfully produced in spite of random errors
in the initial conditions.Comment: 19 pages, 9 figure
Rigorous Numerical Verification of Uniqueness and Smoothness in a Surface Growth Model
Based on numerical data and a-posteriori analysis we verify rigorously the
uniqueness and smoothness of global solutions to a scalar surface growth model
with striking similarities to the 3D Navier--Stokes equations, for certain
initial data for which analytical approaches fail. The key point is the
derivation of a scalar ODE controlling the norm of the solution, whose
coefficients depend on the numerical data. Instead of solving this ODE
explicitly, we explore three different numerical methods that provide rigorous
upper bounds for its solutio
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms
We consider the identification of nonlinear diffusion coefficients of the
form or in quasi-linear parabolic and elliptic equations.
Uniqueness for this inverse problem is established under very general
assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof
of our main result relies on the construction of a series of appropriate
Dirichlet data and test functions with a particular singular behavior at the
boundary. This allows us to localize the analysis and to separate the principal
part of the equation from the remaining terms. We therefore do not require
specific knowledge of lower order terms or initial data which allows to apply
our results to a variety of applications. This is illustrated by discussing
some typical examples in detail
Rigorous a-posteriori analysis using numerical eigenvalue bounds in a surface growth model
In order to prove numerically the global existence and uniqueness of smooth
solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori
upper bounds on the supremum of the numerical range of the linearized operator.
These bounds also have to be easily computable in order to be applicable to our
rigorous a-posteriori methods, as we use them in each time-step of the
numerical discretization. The final goal is to establish global bounds on
smooth local solutions, which then establish global uniqueness.Comment: 19 pages, 9 figure
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