## 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
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