Numerical diagnosis of blow-up of solutions of pseudoparabolic equations

et al. 2018

The initial‐boundary value problem for an equation of ion sound waves in plasma is considered. A theorem on nonextendable solution is proved. The blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analysed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately.

Section: Numerical Localization Of Blow‐upmentioning

confidence: 99%

r_{t}^{1} = r_{x}^{2}

[for more explanations, see Kalitkin et al]).…”

Section: Numerical Localization Of Blow‐upmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the blow‐up phenomena for a 1‐dimensional equation of ion sound waves in a plasma: Analytical and numerical investigation

et al. 2018

“…For numerical solving of this implicit system of ODEs (), we use Rosenbrock scheme with complex coefficient (CROS1) that is the best choice for solving such kind of problems because of its order of accuracy ( O ( τ 1 )), monotonicity, and stability . It is very important to mention that Rosenbrock scheme with complex coefficient would have order of accuracy O ( τ 2 ) in the case of constant matrix M . The implicit system of ODEs with such constant matrix can be obtained if we introduce additional auxiliary variable v = u x x − e ε u in ().…”

Section: Numerical Localization Of Blow‐upmentioning

confidence: 99%

“…In this case, the nodes of the start grid are those of all subsequent grids. At these points, we estimate (see, e.g., ) the error:

{normalΔ}^{(r_{x} N, r_{t} M)} (x, t) = \frac{u^{(r_{x} N, r_{t} M)} (x, t) - u^{(N, M)} (x, t)}{{r_{t}}^{1} - 1} + o ((), τ^{1} + h^{2})

and the effective accuracy order:

p^{e f f} (x, t) = {normal l o g}_{r_{t}} \frac{u^{(r_{x} N, r_{t} M)} (x, t) - u^{(N, M)} (x, t)}{u^{((), r_{x}^{2} N, r_{t}^{2} M)} (x, t) - u^{(r_{x} N, r_{t} M)} (x, t)} .

At each points ( x , t ), where

p^{e f f} (x, t) \to_{N, M \to \infty}^{} p^{t h e o r} = 1,

the solution of the problem has bounded first derivatives in time and second derivative in the spatial variables, and the error estimate is asymptotically sharp as N , M → ∞ . The violation of convergence () indicates the loss of smoothness of the exact solution.…”

Section: Numerical Localization Of Blow‐upmentioning

confidence: 99%

Blow‐up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis

et al. 2016

The initial‐boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analyzed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

Blow‐up for Joseph–Egri equation: Theoretical approach and numerical analysis

2020

This work develops the theory of the blow-up phenomena for Joseph-Egri equation. The existence of the nonextendable solution of two initial-boundary value problems (on a segment and a half-line) is demonstrated. Sufficient conditions of the finite-time blow-up of these solutions, as well as the analytical estimates of the blow-up time, are obtained. A numerical method that allows to precise the blow-up moment for specified initial data is proposed.