1. Numerical methods for modeling of electrophysical problems
Aim of work: an overview of the known methods of numerical calculation of partial
differential equations to problems in electrophysics.
Tasks:
1. Classification of partial differential equation.
2. Electrostatic field and it properties. Elliptic Poisson's equation.
3. Methods of calculation for elliptic equation
4. Propagation of electromagnetic waves in a transmission line
5. Methods of calculation for hyperbolic equation
6. Equations describing propagation of Glow discharge
7. Electron - hole plasmas in SOS-diode
8. Electrical conductor explosion

2. Classification of second-order partial differential equations2
Hyperbolic equation
wave propagation
Parabolic equation
diffusion, heat conductivity,
viscosity
Elliptic equation
stationary potential distribution
of electrostatic field

3. Five-point stencil for calculation of the Laplace equation3
Five-point stencil for calculation of the Laplace equation
cross scheme
diagonal scheme
Laplace equation
The second derivatives are replaced
central discrete analogue
where i, j – node number on the grid

4. Example of systems equation for sixteen unknown nodes of gird 4
Equations system
Matrix of equation coefficients
Column of matrix unknown nodes
Column of matrix known parameters X =

5. Gaussian elimination algorithm for calculation of matrix 5
k=0; i=0;
while( k < size-1 ) {
s=k+1;
while( s < size )
if( matrix[s][i] != 0 )
coef = matrix[k][i]/matrix[s][i]; // коэффициент умножения
while( i < size )
matrix[s][i] = matrix[s][i]*coef - matrix[k][i];
i++; }
i=k;
second_member[s] = coef*second_member[s] - second_member[k];
}//if
s++;
} //while
k++; i=k;
Finding the resulting values
result[k] = second_member[k]/ matrix[k][k];
k--;
while( k < size && k >= 0 ) {
i=k+1;
mix=0;
while( i < size )
mix = mix + matrix[k][i]*result[i];
i++; }
result[k] = (second_member[k]-mix)/matrix[k][k];
Program code is written in C language

6. Gauss-Seidel iterative method6
Gauss-Seidel iterative method
Iterative approximation
if ∆x= ∆y
where n – number of iteration
s=0;
while( s < n ) // number of iterative cycles {
k=1;
while( k < Size_B-1 )
i=1;
while( i < Size_A-1 )
if( voltage[k][i] == 1 ) // coordination array
result[k][i] = 0.25 * (result[k][i+1] + result[k][i-1] + result[k+1][i] + result[k-1][i]);
i++; }
k++;
s++;

7. 7
A Computational region of Laplace's equation and parameters of electrodes
Multi-rod electrode disposed within the closed region
Geometric parameters of electrodes system:
ℓ - electrode length;
d – interelectrode distance;
b – distance between the rods;
h – rod height;
R – rod thickness;
n – number of rod.
Calculation of field strength:
Reducing the grid spacing in 2 times:

8. Distribution of equal potential lines in multi-rod electrode 8
The system of electrodes with different spacing between the rods
0,07m, grid spacing - 1mm
0,07m, grid spacing – 0.25mm
0,035m, grid spacing - 1mm
0,035m, grid spacing – 0.25mm
Common parameters of electrodes: ℓ=0,31 m, d=0,07 m, h=0,05 m, R=2 mm, n=5

9. 9
Force lines of electrostatic field at plane-parallel electrode geometry

10. Poisson's equation for an inhomogeneous medium10
Poisson's equation for an inhomogeneous medium
ρ(x, y) - density distribution of free charge
ε(x, y) - absolute dielectric permeability
Finite - different record of Poisson's equation
Expression for the central node

11. Dielectric plate between plane-parallel electrodes11
Dielectric plate between plane-parallel electrodes
φ2 > φ1, ε0 = 1, ε1 = 5, ρ1 = 0 μQ/m3
φ2 > φ1, ε0 = ε1 = 1, ρ1 =  0.5 μQ/m3
φ2 > φ1, ε0 = ε1 = 1
potential surface
ρ1 = 1 μQ/m3 , ρ2 = 0.5 μQ/m3, ρ3 = 0 μQ/m3 ,
ρ4 =  0.5 μQ/m3 , ρ5 =  1 μQ/m3
ρ1 = 1 μQ/m3

12. equivalent circuit of transmission line12
Propagation of electromagnetic wave along the transmission line
equivalent circuit of transmission line
transmission line
Hyperbolic equations for
transmission line with losses
Hyperbolic equations for
transmission line without losses
Distributed parameters of line
C0 – capacitance per unit length,
L0 – inductance per unit length,
R0 – resistance per unit length,
G0 – leakage conductance.
– are voltage and current on line,
that are functions of position and time

13. Method of calculation hyperbolic equation13
Method of calculation hyperbolic equation
Conservative method of Lax
Lax scheme
Lax-Wendroff scheme
Two-step Lax – Wendroff method
First (Lax) steps:
Second step:

14. Reflection of the wave from the end of the transmission line14
Reflection of the wave from the end of the transmission line
Open end of the line
Boundary conditions
Umax = 10 kV,
tmax = 2 ns
C0 = 20 pF, L0 = 50 nH
Distribution of voltage along of line
Distribution of current along of line
V, kV
I, A
Time - 20 ns
Time - 20 ns

15. Reflection of the wave from the end of the transmission line15
Reflection of the wave from the end of the transmission line
Closed end of the line
Boundary conditions
Umax = 10 kV,
tmax = 2 ns
C0 = 20 pF, L0 = 50 nH
Distribution of voltage along of line
Distribution of current along of line
V, kV
I, A
Time - 20 ns
Time - 20 ns

16. Equations describing propagation of a Glow discharge16
Equations describing propagation of a Glow discharge
Initial and boundary conditions
Differential equation and ionization coefficient

17. Finite-different record of equations17
Initial and boundary conditions
Equation are written according to the Lax schema

18. Electron-hole plasmas in SOS-diode18
Electron-hole plasmas in SOS-diode
Fundamental system of equations
Continuity equations for carriers of charge
Flows electrons and holes
Poison’s equation
Field intensity equation
Heat equation

19. Coefficients mobility, ionization and recombination19
Coefficients mobility, ionization and recombination

20. Electrical conductor explosion20
Electrical conductor explosion
Electromagnetic equations
Continuity equations for density of liquid metal
Fluid flow equation
Heat equation
Equations of state and electrical conductivity

21. The presented materials are still being refined…21
The presented materials are still being refined…