×

Finite difference approximations for nonlinear first order partial differential equations. (English) Zbl 1041.65067

The authors consider the following first-order partial differential equation \[ \partial_t z(t,x)= f(t, x,zH,x),\;\partial_x z(t, x))\tag{1} \] with initial condition \[ z(0,x)= \varphi(x),\quad x\in [-b,b].\tag{2} \] Here \(t\in\mathbb R\), \(x= (x_1,\dots, x_n)\in \mathbb R^n\), \(E\) is the Haar pyramid \(\{(t, x)\in \mathbb R^{1+ n}: t\in [0,a], -b+ Mt\leq x\leq b- Mt\}\), \(f:E\times \mathbb R\times \mathbb R^n\to \mathbb R\), \(a> 0\), \(M= (M_1,\dots, M_n)\in \mathbb R^n_+\), \(\mathbb R_+= [0,\infty)\), \(b= (b_1,\dots, b_n)\in \mathbb R^n\), \(Ma\leq b\), \(\partial_x z=(\partial_{x_1}z,\dots, \partial_{x_n}z)\), \(\varphi: [-b,b]\to \mathbb R\). (Vector inequalities above denote those between the corresponding components of vectors.)
After introducing an additional function \(u= \partial_x z\) the authors linearize in some manner the equation (1). Together with equations received by differentiating (1) with respect to \(x_r\), \(1\leq r\leq n\), they obtain a differential system for the vector function \(u\) subjected to (2) and \(u(0, x)= \partial_x \varphi(x)\), \(x\in [-b,b]\), Next, the authors approximate the differential problem by standard finite difference methods, prove their convergence and obtain error estimates (of course under suitable assumptions on \(f\), \(\varphi\) and the steps of mesh).
A numerical example for \(n=1\), \(E= \{(t, x)\in \mathbb R^2L t\in [0,1]\), \(| x|\leq 2-2t\}\), \(\partial_t z(t,x)= {1\over 2}\sin(1+ \partial_x z(t, x))+ 1+ x^3- {1\over 2}\sin(1+ 3x^2 t)\), \(z(0, x)= 0\), \(x\in [-2,2]\) is examined.
Reviewer: S. Burys (Kraków)

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
PDF BibTeX XML Cite
Full Text: EuDML