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Numerical solution of a second-order non-linear PDE

2

3

$begingroup$

I'm interested in solving the following non-linear terminal value problem
$$
0 = frac{partial u}{partial t} + frac{1}{2}sigma^2x
frac{partial^2 u}{partial x^2}
+ lambda(x_0 - x)frac{partial u}{partial x} - phi
+ frac{1}{x}u^2,qquad u(T, x) = -alpha.
$$
on $(t, x)in(0, T)times(0, infty)$, where $sigma>0$, $lambda>0$, $x_0>0$, $phi>0$ are constants. For context, this PDE arose from an application of the Hamilton-Jacobi-Bellman equation to a stochastic control problem that I'm studying.

I think it's unlikely that an analytic solution is available, and I expect I'll need to use numerical methods to examine the solution. My first thought is to use a finite difference scheme, such as Crank-Nicolson, but I'm not sure of a suitable boundary condition to use on the solution grid for large $x$.

I'd be very grateful for any suggestions on how to proceed. I'm not very familiar with the literature on the numerical solution of PDEs, so even pointers towards possible references would be really helpful.

pde numerical-methods

share|cite|improve this question

asked Mar 22 at 13:25

Alex WAlex W

1427

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I’d first think whether the pde problem is well-posed. Is it?
  $endgroup$
  – VorKir
  Mar 30 at 5:01
  

  
  
  
  

add a comment | 

2

3

$begingroup$

I'm interested in solving the following non-linear terminal value problem
$$
0 = frac{partial u}{partial t} + frac{1}{2}sigma^2x
frac{partial^2 u}{partial x^2}
+ lambda(x_0 - x)frac{partial u}{partial x} - phi
+ frac{1}{x}u^2,qquad u(T, x) = -alpha.
$$
on $(t, x)in(0, T)times(0, infty)$, where $sigma>0$, $lambda>0$, $x_0>0$, $phi>0$ are constants. For context, this PDE arose from an application of the Hamilton-Jacobi-Bellman equation to a stochastic control problem that I'm studying.

I think it's unlikely that an analytic solution is available, and I expect I'll need to use numerical methods to examine the solution. My first thought is to use a finite difference scheme, such as Crank-Nicolson, but I'm not sure of a suitable boundary condition to use on the solution grid for large $x$.

I'd be very grateful for any suggestions on how to proceed. I'm not very familiar with the literature on the numerical solution of PDEs, so even pointers towards possible references would be really helpful.

pde numerical-methods

share|cite|improve this question

asked Mar 22 at 13:25

Alex WAlex W

1427

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I’d first think whether the pde problem is well-posed. Is it?
  $endgroup$
  – VorKir
  Mar 30 at 5:01
  

  
  
  
  

add a comment | 

$begingroup$

I'm interested in solving the following non-linear terminal value problem
$$
0 = frac{partial u}{partial t} + frac{1}{2}sigma^2x
frac{partial^2 u}{partial x^2}
+ lambda(x_0 - x)frac{partial u}{partial x} - phi
+ frac{1}{x}u^2,qquad u(T, x) = -alpha.
$$
on $(t, x)in(0, T)times(0, infty)$, where $sigma>0$, $lambda>0$, $x_0>0$, $phi>0$ are constants. For context, this PDE arose from an application of the Hamilton-Jacobi-Bellman equation to a stochastic control problem that I'm studying.

I think it's unlikely that an analytic solution is available, and I expect I'll need to use numerical methods to examine the solution. My first thought is to use a finite difference scheme, such as Crank-Nicolson, but I'm not sure of a suitable boundary condition to use on the solution grid for large $x$.

I'd be very grateful for any suggestions on how to proceed. I'm not very familiar with the literature on the numerical solution of PDEs, so even pointers towards possible references would be really helpful.

pde numerical-methods

share|cite|improve this question

asked Mar 22 at 13:25

Alex WAlex W

1427

$endgroup$

share|cite|improve this question

asked Mar 22 at 13:25

Alex WAlex W

1427

share|cite|improve this question

asked Mar 22 at 13:25

Alex WAlex W

1427

asked Mar 22 at 13:25

Alex WAlex W

1427

asked Mar 22 at 13:25

Alex WAlex W

1427