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difference analog of equation 3.72 as average to zero after integration over 6. The time derivative of fo is given by the 0-th Only fo remains under the above integral, all the rest of the expansion coefficients To evaluate the left side of this equation we need a representation for (v?);41 yo. W calculate the number of particles inside Dy with formula 3.69:

Table 8 difference analog of equation 3.72 as average to zero after integration over 6. The time derivative of fo is given by the 0-th Only fo remains under the above integral, all the rest of the expansion coefficients To evaluate the left side of this equation we need a representation for (v?);41 yo. W calculate the number of particles inside Dy with formula 3.69:

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Notations
Notations
Cosine of the polar angle @ in the velocity space
Cosine of the polar angle @ in the velocity space
Figure 1-1: A pair of waves produced by the transformation of the advection equation to a half-line.
Figure 1-1: A pair of waves produced by the transformation of the advection equation to a half-line.
In the Py approximation, the system of equations for the expansion coefficients f,,
In the Py approximation, the system of equations for the expansion coefficients f,,
equation 2.11 (or 2.12) and simplifying (but not dropping anything), we obtain
equation 2.11 (or 2.12) and simplifying (but not dropping anything), we obtain
The expansion coefficients in formula 2.86 can be found as where we used summation over 0 < q and full range of ¢-s (see formulas 2.70 and
The expansion coefficients in formula 2.86 can be found as where we used summation over 0 < q and full range of ¢-s (see formulas 2.70 and
The above derivation uses a component form of formula D.26 for e“”. The result of component form of equation 2.97 yield the derivation can be conveniently summarized in matrix form
The above derivation uses a component form of formula D.26 for e“”. The result of component form of equation 2.97 yield the derivation can be conveniently summarized in matrix form
are represented by thick colored lines while the rest of the expansion coefficients f,s7 are indicated by thin black lines. On the right are the graphs of the corresponding
are represented by thick colored lines while the rest of the expansion coefficients f,s7 are indicated by thin black lines. On the right are the graphs of the corresponding
Figure 2-2: Analytic solution for Pe, at t=0.1, 0.2, 0.3, 0.4.
Figure 2-2: Analytic solution for Pe, at t=0.1, 0.2, 0.3, 0.4.
Figure 3-1: Absolute values of eigenvalues of matrix B. figure 3-1. The line max(b;,) + 0.82(N — 1) +0.0024(N — 1)? shows the growth of the largest absolute eigenvalue of matrix B for 1 < N < 31.
Figure 3-1: Absolute values of eigenvalues of matrix B. figure 3-1. The line max(b;,) + 0.82(N — 1) +0.0024(N — 1)? shows the growth of the largest absolute eigenvalue of matrix B for 1 < N < 31.
It can be verified that all the particles that were initially present and later injected into the interval remain there afterwards. The described model can be implemented
It can be verified that all the particles that were initially present and later injected into the interval remain there afterwards. The described model can be implemented
cally illustrated in figure 3-2. The length of the arrows and their direction correspond Figure 3-2: Behavior of the characteristic variables gz near zero for Py and 0 < ay. [he behavior of the characteristic variables given by the above formula is graphi- to the absolute values and signs of the eigenvalues of matrix A. Arrows show the
cally illustrated in figure 3-2. The length of the arrows and their direction correspond Figure 3-2: Behavior of the characteristic variables gz near zero for Py and 0 < ay. [he behavior of the characteristic variables given by the above formula is graphi- to the absolute values and signs of the eigenvalues of matrix A. Arrows show the
Figure 3-3: Conical region in the velocity space and sectors on the v,0v, plane defined by the functions a?’ (6). in figure 3-3. The conical region defined by the function a} (6) in the velocity space
Figure 3-3: Conical region in the velocity space and sectors on the v,0v, plane defined by the functions a?’ (6). in figure 3-3. The conical region defined by the function a} (6) in the velocity space
approximately describes the action of R**’(a,7) and has the following equation for its characteristic curve on the (t, #)-plane difference analog of the analytical solution 3.23 of equation 3.5. Solutions 3.30 and
approximately describes the action of R**’(a,7) and has the following equation for its characteristic curve on the (t, #)-plane difference analog of the analytical solution 3.23 of equation 3.5. Solutions 3.30 and
Figure 3-4: On the left, infinitesimal motions along the characteristic curves for solutions 3.62 and 3.66. On the right, action of operators 7,2¢"(a,7) and R2*(a,7). and approximate actions of the operators 7,2¢’(a,r) and R**’(a,T). As one woul
Figure 3-4: On the left, infinitesimal motions along the characteristic curves for solutions 3.62 and 3.66. On the right, action of operators 7,2¢"(a,7) and R2*(a,7). and approximate actions of the operators 7,2¢’(a,r) and R**’(a,T). As one woul
(see also formula 3.76) are the finite-difference equivalents of the spatial flux and is multiplied by 47(v?)j41 yov and after summation over the velocity mesh intervals particle density. Note that formula 3.138 defines the spatial flux J; at the edge of
(see also formula 3.76) are the finite-difference equivalents of the spatial flux and is multiplied by 47(v?)j41 yov and after summation over the velocity mesh intervals particle density. Note that formula 3.138 defines the spatial flux J; at the edge of
Two important features have to be noted. First, though the secondary electron Using the above orthogonality relation, the expansion coefficients can be constructed 3.154 and 3.161, equation 3.152 is satisfied exactly at the left and right boundaries by
Two important features have to be noted. First, though the secondary electron Using the above orthogonality relation, the expansion coefficients can be constructed 3.154 and 3.161, equation 3.152 is satisfied exactly at the left and right boundaries by
Figure 3-6: Analytical and numerical solutions for P7 at t=0.5, 0.6, 0.7, 0.8.
Figure 3-6: Analytical and numerical solutions for P7 at t=0.5, 0.6, 0.7, 0.8.
Figure 3-8: Analytical and numerical solutions for Pg at t=0.5, 0.6, 0.7, 0.8.
Figure 3-8: Analytical and numerical solutions for Pg at t=0.5, 0.6, 0.7, 0.8.
Figure 3-9: Distribution of scattered electrons in the excitation collisions.
Figure 3-9: Distribution of scattered electrons in the excitation collisions.
Figure 3-10: Distribution of scattered and produced electrons in the ionization colli- sions.
Figure 3-10: Distribution of scattered and produced electrons in the ionization colli- sions.
Figure 3-11: Cross-sections of elastic, excitation and ionization collision processes for argon.
Figure 3-11: Cross-sections of elastic, excitation and ionization collision processes for argon.
Figure 3-12: Stationary solution for E = 1000 Td and P;, ionization as excitation.
Figure 3-12: Stationary solution for E = 1000 Td and P;, ionization as excitation.
Figure 3-13: Stationary solution for FE = 1000 Td and Pys, realistic ionization.
Figure 3-13: Stationary solution for FE = 1000 Td and Pys, realistic ionization.
Figure 3-14: Stationary solution for FE = 100 Td and Pj5, realistic ionization
Figure 3-14: Stationary solution for FE = 100 Td and Pj5, realistic ionization
Figure 3-16: Parallel implementation of B(r).
Figure 3-16: Parallel implementation of B(r).
Figure 3-15: Left semi-reflecting boundary.
Figure 3-15: Left semi-reflecting boundary.
Figure 4-1: Two plane-electrode geometry. field is approximately described by
Figure 4-1: Two plane-electrode geometry. field is approximately described by
Figure 4-2: Townsend’s ionization coefficient for Ar. applied voltage of U = 200 V. The dependence of the discharge current on time found
Figure 4-2: Townsend’s ionization coefficient for Ar. applied voltage of U = 200 V. The dependence of the discharge current on time found
Figure 4-3: Dependence of the discharge current on time. of the distribution functions in the figures follows the order of the points selected
Figure 4-3: Dependence of the discharge current on time. of the distribution functions in the figures follows the order of the points selected
Figure 4-5: Electron distribution function at 60 ns.
Figure 4-5: Electron distribution function at 60 ns.
Figure 4-6: Electron distribution function at 60 ns (continued).
Figure 4-6: Electron distribution function at 60 ns (continued).
Figure 4-9: Electron distribution function at 117 ns.
Figure 4-9: Electron distribution function at 117 ns.
Figure 4-10: Electron distribution function at 117 ns (continued).
Figure 4-10: Electron distribution function at 117 ns (continued).
Figure 4-11: Electric field, densities of electrons and ions, electron drift velocity and mean electron energy at 200 ns.
Figure 4-11: Electric field, densities of electrons and ions, electron drift velocity and mean electron energy at 200 ns.
Figure 4-12: Electron distribution function at 200 ns.
Figure 4-12: Electron distribution function at 200 ns.
Figure 4-13: Electron distribution function at 200 ns (continued).
Figure 4-13: Electron distribution function at 200 ns (continued).
Figure A-1: Spherical coordinates in the velocity space.
Figure A-1: Spherical coordinates in the velocity space.
where integration in the surface integrals goes over the boundary regions: Ds, and surface integrals by the divergence theorem For later we assume that extension of the integration region in the velocity space
where integration in the surface integrals goes over the boundary regions: Ds, and surface integrals by the divergence theorem For later we assume that extension of the integration region in the velocity space
where matrix A is defined by formula 2.5. Rewriting the previous equation for the
where matrix A is defined by formula 2.5. Rewriting the previous equation for the
relations for the Bessel functions 2.13, we have to satisfy equations
relations for the Bessel functions 2.13, we have to satisfy equations
the arguments of the spherical Bessel functions positive, the expressions that follow by formulas ] H.15 and | 1.16, we obtain the second equality signs are transformed into the expressions following the third q@; and a, satisfying @;A = Q@;a,, and using the properties of the operator L*(v) given quality signs by using the parity properties of the spherical Bessel functions (see Introducing the adjoint eigensystem of matrix A (discussed in Appendix D), that is
the arguments of the spherical Bessel functions positive, the expressions that follow by formulas ] H.15 and | 1.16, we obtain the second equality signs are transformed into the expressions following the third q@; and a, satisfying @;A = Q@;a,, and using the properties of the operator L*(v) given quality signs by using the parity properties of the spherical Bessel functions (see Introducing the adjoint eigensystem of matrix A (discussed in Appendix D), that is
Figure F-1: The solid circle displays azimuthally symmetric points. duced through rotation by angle ¢. Azimuthally symmetric points (Fig. F-1) form a circle with its center positioned on axis Ov, at the point corresponding to coordi- For a given point v in the velocity space, azimuthally symmetric points are pro-
Figure F-1: The solid circle displays azimuthally symmetric points. duced through rotation by angle ¢. Azimuthally symmetric points (Fig. F-1) form a circle with its center positioned on axis Ov, at the point corresponding to coordi- For a given point v in the velocity space, azimuthally symmetric points are pro-
Figure F-2: Azimuthally symmetric density function on the left and particles dis- tributed in the velocity space according to this density function on the right. bution function can be displayed on the v,Ov, plane as in Fig. F-2.
Figure F-2: Azimuthally symmetric density function on the left and particles dis- tributed in the velocity space according to this density function on the right. bution function can be displayed on the v,Ov, plane as in Fig. F-2.
Figure H-1: Scattering from dv’ into dv.
Figure H-1: Scattering from dv’ into dv.
Figure H-2: Angle @,,., is measured from a projection of axes Ov, to a projection of vector v’ on a plane perpendicular to vector v. Introducing angle ¢,,,. (see Fig. H-2), we can utilize the addition theorem for associ-
Figure H-2: Angle @,,., is measured from a projection of axes Ov, to a projection of vector v’ on a plane perpendicular to vector v. Introducing angle ¢,,,. (see Fig. H-2), we can utilize the addition theorem for associ-
Jsing formulas H.29, H.31 and substitution H.30, we can see that normalization electrons, n, are scattered and we find dn = n4rv7dup,(v, v) of the electrons of type 1 If electron 1 is scattered into a velocity interval [v,v + dv], electron 2 is found in
Jsing formulas H.29, H.31 and substitution H.30, we can see that normalization electrons, n, are scattered and we find dn = n4rv7dup,(v, v) of the electrons of type 1 If electron 1 is scattered into a velocity interval [v,v + dv], electron 2 is found in
tion o(v?, v2), conservation of electron density When both the scattered and produced electrons have the same density distribu-
tion o(v?, v2), conservation of electron density When both the scattered and produced electrons have the same density distribu-
Related topics:
Boltzmann equationNumerical Methods for hyperbolic PDE
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