Abstract: According to the stochastic differential equation of reservoir water level process during flood discharge, the reservoir water level and its waves under random interference are simulated by numerical solution method.

The probability of flood overtopping the dam and the sample average value of reservoir water level process at different times are calculated by corresponding formulas, and compared

Under the random interference of the same intensity, it is of great guiding significance to determine the advantages and disadvantages of various flood discharge schemes.

Meaning.

Keywords: stochastic differential equation; Numerical solution; Euler method; Flood discharge risk

1 quotation

Date of receipt: June 27, 2005

Fund Project: National Natural Science Foundation of China (60474037); Support Plan for Excellent Talents in the New Century of Ministry of Education (NCET-04-4 15)

Risk analysis is an extremely effective tool to deal with natural disasters such as floods and storm surges.

Because of the complexity of stochastic differential equations, except for some linear or special structure equations, we can find obvious

Few stochastic differential equations show solutions. The stochastic differential equation discussed in this paper does not have the above properties, so it cannot be solved.

Jiang satisfies the Fokker-plank forward equation according to the first-order probability density function of his solution process, and here

Cheng is also a partial differential equation, so the numerical solution of partial differential equation can be obtained by finite difference method [6], but this method can't.

The probability characteristics are obtained, so JC calculation method is used to approximately calculate the probability of flood passing through the dam crest [7]. It is not difficult to see that this kind of

Because of many transformations, the error of the method is relatively large.

In this paper, the numerical solution method of stochastic differential equation is used to analyze and summarize the interference of reservoir water level in Brownian motion with examples

Random fluctuations under; The risk probability of flood overflow and the mathematical expectation of reservoir water level process at different times are directly obtained.

In addition, different schemes are analyzed and compared to determine which scheme has better effect, thus improving the flood control decision-making process.

Provide a certain basis.

Stochastic differential equation of flood dispatching process

In the process of flood dispatching, inflow flood and outflow flood are stochastic processes, and the water level satisfies stochastic differential equation [6]:

dH(t) =Q-(t) -q-(H，c)G(H)dt+dB(t)G(H)

H(t0) =H0

( 1)

H(t) is the process of reservoir water level; H0 is the initial water level of the reservoir, which is a random variable; Q(t) is the flood inflow at any moment.

Water quantity; Q(h, c) is the flood discharge at the corresponding time; Q-, q- are the average hydrograph of incoming flow and flood discharge respectively; C is the flow coefficient.

Equal hydraulic parameters. G(H) =dW(H)dH, W(H) is the reservoir capacity, and B(t) is the Wiener transition with zero mean value.

Cheng, dB(t)/dt is normal white noise, and the one-dimensional probability density function f(B) of B(t) is:

f(B) = 1

2πt σexp -B22σ2t。

As can be seen from the above formula, E[B(t)] = 0 and D[B(t)] =σ2t. The risk rate of flood discharge at dam crest is defined as Pf=

Pf[H Z], where z is the corresponding dam height.

3 calculation method

Because the explicit solution of stochastic differential equation is rarely found, its numerical solution method has been widely studied and applied.

In the numerical method of ordinary differential equation, the numerical method of stochastic differential equation introduces random increment, which will consider time.

The interval is limited, and the approximate value of the sample orbit is gradually generated at the node. The numerical solution methods mainly include: Eu-

Le method, milstein method, Runge-Kutta method, etc. Euler method is used here.

3. 1 Euler approximation method for solutions of stochastic differential equations

Consider general stochastic differential equations:

dXt=a(t，Xt)dt+b(t，Xt)dWt(2)

Where t0 t T, the initial condition is Xt0=X0. We discretize the time interval [t0, T]:

t0 =τ0 & lt； τ 1 & lt； …& lt； τn & lt； …& lt； τN=T。

Using Euler approximation method [8], a continuous process Y= {Y(t), t0 T} satisfies the following iterative format:

Yn+ 1=Yn+a(τn，Yn)(τn+ 1-τn) +b(τn，Yn)(Wτn+ 1-Wτn)

Where n = 0, 1, 2, …, n- 1, y0 = x0. Let the finite discrete random variable obtained by step-by-step iteration be

Approximate solutions of original stochastic differential equations at corresponding time nodes. Obviously, if the diffusion coefficient is zero, the original stochastic differential equation

Degenerate to ordinary differential equations, so the Euler method of stochastic differential equations degenerates to the Euler method of ordinary differential equations.

As far as numerical methods are concerned, their strong convergence is generally discussed.

Definition 1[8] For the discrete approximation sequence Yδ with maximum step length δ, it strongly converges to an Ito∧ at time t.

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