Numerical Study of Interacting Particles Approximations

Objective: to develop a numerical method based on the interacting particles approximation (propagation of chaos) for the solution of a large class of evolution problems involving the fractional Laplacian operator and a nonlocal quadratic-type nonlinearity.

Methodology: Coupled stochastic differential equations driven by Lévy symmetric $\alpha$-stable processes are integrated numerically using Euler's method and the solutions of the governing equations are obtained from their statistics.

1. Model Problems and Applications

   (PDF)$$\qquad\left\{\begin{array}{ll} \partial_{t} u=\sigma^{2} \Delta_{\alpha} u+ \nabla\cdot (uB(u)) \\ u(x,0)=u_{0}(x) \end{array} \right.$$

   (CDF)$$\qquad\left\{\begin{array}{ll} \partial_{t} v=\sigma^{2} \Delta_{\alpha} v+\nabla v \cdot B(v)\\ v(x,0)=v_{0}(x) \end{array} \right.$$

where $u,\: v:\Omega \times [0,T] \subset \mathbb{R}^{d} \times \mathbb{R}^{+} \rightarrow \mathbb{R}$, $\sigma >0$, for $0 < \alpha \leq 2$,
$\Delta_{\alpha} := -(-\Delta)^{\alpha/2}$ is the fractional (power of the) Laplacian in $\mathbb{R}^{d}$ defined via the Fourier transform $\mathcal{F}$:

$$\mathcal{F}(\Delta_{\alpha} u)(\omega) := -(\vert\omega\vert^{2})^{\alpha/2} (\mathcal{F}u)(\omega),\qquad \omega \in \mathbb{R}^{d},$$

and $B(u)$ is a linear $\mathbb{R}^{d}$-valued integral operator with the kernel $b:\mathbb{R}^{d}\times \mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$:

$$B(u)(x) = \int_{\mathbb{R}^{d}} b(x-y)u(y) dy$$

$$\vert b(z)\vert \leq c\vert z\vert^{\beta-d}$$

for some $0<\beta<d$ and $0<c$.

"The study of one individual gives information on the behavior of the group."

$$\fbox{PDE}$$

$$\partial_{t}u = \sigma^{2} \Delta_{\alpha}u+\nabla\cdot\left(u\int_{\mathbb{R}^{d}} b(x-y)u(y) dy\right)$$

$$\fbox{SDE}$$

$$dX_{t} = \sigma dS_{t}-\left(\int_{\mathbb{R}^{d}} b(X_{t}-y)u(y) dy\right) dt$$

$$\fbox{N-Particle System}$$

$$dX^{i}_{t} = \sigma\,dS^{i}_{t}-\frac{1}{N}\sum_{j\neq i} b_{\epsilon}(X^{i}_{t}-X^{j}_{t})\,dt$$

$$\fbox{Discrete System}$$

$$X^{i}_{k} = X^{i}_{k-1}+\sigma (S^{i}_{k}-S^{i}_{k-1}) -\frac{1}{N}\sum_{j\neq i} b_{\epsilon}(X^{i}_{k-1}-X^{j}_{k-1}) \Delta t.$$

1. Normal Diffusion ($\alpha =2$)
   • $b(x,y)=c(x-y)\vert x-y\vert^{-d}$
     1. $c<0$ : Eq.(PDF) models the diffusion of charge carriers in electrolytes, semiconductors or plasmas interacting via Coulomb forces.
     2. $c>0$ : Eq.(PDF) describes gravitational interaction of particles in a cloud, or galaxies in a nebula.
   • $b(x,y)=(2 \pi)^{-1}(x_{2}-y_{2},y_{1}-x_{1})\vert x-y\vert^{-2}$ in $\mathbb{R}^{2}$
     1. $b(x,y)$ is the Biot-Savart kernel.
     2. Eq.(PDF) is equivalent to the vorticity formulation of the Navier-Stokes equations.
   • $b(x,y)=c\delta (x-y)$ in $\mathbb{R}$
     Eq.(PDF) becomes the classical Burgers equation for $u$
     

     $$\partial_{t} u=\sigma^{2} \partial_{xx}u+c \partial_{x}(u^{2}).$$

     
     
   • $b(x,y)=c H (x-y)$ in $\mathbb{R}$
     Eq.(CDF) becomes the classical Burgers equation for $v$
     

     $$\partial_{t} v=\sigma^{2} \partial_{xx}v+c v\partial_{x}v.$$

     
     Heaviside function $H(z)$ is defined by
     

2. Anomalous Diffusion ( $0<\alpha<2$)
   In the physical literature such fractal diffusions have been vigorously studied in the context of statistical mechanics, hydrodynamics, acoustics, relaxation phenomena and biology. They also appear in nonlinear models of interfacial growth which involve hopping and trapping effects.

• $b(x,y)=(c_{1}\delta(x-y),c_{2}\delta(x-y),\cdots,c_{d}\delta(x-y))$ in $\mathbb{R}^{d}$
  Eq.(PDF) becomes the fractal Burgers equation
  

  $$\partial_{t} u=\sigma^{2} \Delta_{\alpha} u +c\cdot \nabla (u^{2}),$$

  
  where $c=(c_{1},c_{2},\cdots,c_{d})\in \mathbb{R}^{d}$.
• $b(x,y)=c\delta (x-y)$ in $\mathbb{R}$
  Eq.(CDF) becomes the fractal KPZ equation
  (the Kadar, Parisi and Zhang equation)
  
  

  $$\partial_{t} v=\sigma^{2} \Delta_{\alpha} v +c(\partial_{x}v)^{2}.$$

  
  

2. 1D Fractal Burgers' equation

$$\left\{\begin{array}{ll} \partial_{t} v -v \partial_{x}v= \Delta... ... 0 < \alpha \leq 2\\ v(x,0)=\frac{1}{2}(\tanh(x/4)+1). \end{array} \right.$$

This is an example of (CDF) for taking $b(x-y)=H(x-y)$, $\sigma=1$, and $d=1$ where $H$ is a Heaviside function.

$$\left\{\begin{array}{ll} X^{i}_{k}=X^{i}_{k-1}+(S^{i}_{k}-S^{i}_... ...X^{i}_{k-1}-X^{j}_{k-1})\Delta t\\ X^{i}_{0}=x^{i}_{0}. \end{array} \right.$$

$\alpha =2$: the exact solution is a travelling wave solution of the form

$$v(x,t)=\frac{1}{2}(\tanh((x+t/2)/4))+1).$$

$$\qquad\left\{\begin{array}{ll} \partial_{t}u +u\partial_{x}u=\si... ...\alpha \leq 2\ u(x,0)=1/\sqrt{2\pi} \exp(-x^{2}/2) \end{array} \right. .$$

Click here for Particle 12

Click here for Solution 12

vs.

Click here for Particle 34

Click here for Solution 34

3. Fractal Burgers-KPZ Equations

   (KPZ)$$\qquad\left\{\begin{array}{ll} \partial_{t}v+(\partial_{x}v)^{2}... ...uad 0 < \alpha \leq 2\\ v(x,0)=\frac{1}{2}(\tanh(x)+1) \end{array} \right.$$

   (BE)$$\qquad\left\{\begin{array}{ll} \partial_{t}u+\partial_{x}(u^{2})... ...u, \qquad 0 < \alpha \leq 2\\ u(x,0)=1/(2\cosh^{2}(x)) \end{array} \right.$$

$$\left\{\begin{array}{ll} X^{i}_{k}=X^{i}_{k-1}+(S^{i}_{k}-S^{i}_... ...(X^{i}_{k-1}-X^{j}_{k-1})\Delta t\\ X^{i}_{0}=x^{i}_{0} \end{array} \right.$$

 

KPZ		BE

4. Navier-Stokes Equation

$$\qquad\left\{\begin{array}{ll} \partial_{t}u +\int_{\mathbb{R}^{... ...(x_{1},x_{2},1/2)=1/(2\pi) \exp(-(x_{1}^{2}+x_{2}^{2})/2) \end{array} \right.$$

where $u=u(x_{1},x_{2},t)$, $(x_{1},x_{2}) \in \mathbb{R}^{2}$, $t\in [1/2,T]$, and $K$ is the Biot-Savart kernel. The initial condition $u(x_{1},x_{2},t=1/2)$ corresponds to the exact solution

$$u(x_{1},x_{2},t)=\frac{1}{4\pi t}\exp{(-(x_{1}^{2}+x_{2}^{2})/4t)}$$

representing an isolated vortex.

 

Exact (left) and approximate (right) solution to the 2D incompressible Navier-Stokes equation in the vorticity formulation for (a) t=0.5 (b) t=1 (c) t=1.5. $N=120,000$ particles, smoothing parameter $\epsilon=1.0$.

 

5. 2D Scalar Burgers' Equation

$$\qquad\left\{\begin{array}{ll} \partial_{t}u +u\partial_{x_{1}}u... ...x_{1},x_{2},0)=1/(2\pi) \exp(-(x_{1}^{2}+x_{2}^{2})/2) \end{array} \right. .$$

 

Interacting particles approximation to the solution of the 2D fractal ( $\alpha=1.5$) Burgers' equation using 600,000 particles with the smoothing parameter $\epsilon=1.0$. (a) Exact initial condition (b) Approximate initial condition (c) Approximation at t=0.5 (d) Approximation at t=1.

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