Numerical Methods for hyperbolic PDE

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Numerical methods for hyperbolic partial differential equations (PDEs) are computational techniques used to approximate solutions of hyperbolic PDEs, which describe wave propagation and dynamic systems. These methods include finite difference, finite volume, and spectral methods, focusing on stability, accuracy, and convergence to effectively handle the characteristics of hyperbolic equations.

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Application of SOR iteration with nonlocal arithmetic discretization scheme for solving Burger's equation

by Mohd U S R A N Alibubin

2025, AIP Conference Proceedings 2013

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Monte Carlo analysis of field-dependent electron avalanche coefficients in nitrogen at atmospheric pressure

by Ravi Joshi

2023, Physics of Plasmas

Calculations of electron impact ionization of nitrogen gas at atmospheric pressure are presented based on the kinetic Monte Carlo technique. The emphasis is on energy partitioning between primary and secondary electrons, and three... more

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Class of invariants for a time dependent linear potential

by Farid Benamira

2022, Indian Journal of Physics

General solution of the one-dimensional Schrödinger equation in presence of a time-dependent linear potential is reconsidered in the context of Lewis-Riesenfeld and unitary transformation approaches. Three invariant operators are... more

where 4 is an arbitrary real number. Note that the prefactor is introduced here to make eigenfunctions normalized in the sense of Dirac, ie. [°° W(x, t)W,(x, dx = 6(A — 7’). which coincides with the invariant reported earlier [14]. Taking no = 0 and py # 0, the complete set of the solutions of nrohlem are then civen hv nlane-waves as [14]

From Eqs. (21) and (22), we see that the position and momentum uncertainties do not depend on external force f(t). Further, the uncertainty product is given by It is straightforward to calculate the position and momentum uncertainties in state ,,(x, t); they are given by

Thus, for each value v, corresponds to a solution as an Airy wave-packet whose centre evolves along the trajectory given by

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Parallel solution of the discretised and linearised G-heat equation

by Pierre Spiteri

2021, International Journal of High Performance Computing and Networking

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Wishart laws and variance function on homogeneous cones

by Hideyuki Ishi

2021, Probability and Mathematical Statistics

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.

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Legendre Polynomial Expansion of the Electron Boltzmann Equation Applied to the Discharge in Argon

by Yuriy Sosov

2021, Dissertation

The main effort of the present dissertation is to establish a framework for construction of the numerical solution of the system of partial differential equations for the coefficients in the N-term expansion of the solution of the... more

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 3-12: 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 3-14: Stationary solution for FE = 100 Td and Pj5, realistic ionization

Figure 3-16: Parallel implementation of B(r).

Figure 3-15: Left semi-reflecting boundary.

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 4-3: 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 4-6: Electron distribution function at 60 ns (continued).

Figure 4-9: Electron distribution function at 117 ns.

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 4-12: Electron distribution function at 200 ns.

Figure 4-13: Electron distribution function at 200 ns (continued).

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

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 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-

Cosine of the polar angle @ in the velocity space

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

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

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 1-1: A pair of waves produced by the transformation of the advection equation to a half-line.

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

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.

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

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 3-4: 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

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 3-8: 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.

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:

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

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

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-

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A Comparative Application of Jacobi and Gauss Seidel's Numerical Algorithms in Page Rank Analysis.     . Introducing a relaxation parameter λ ≠ 0 leads to a generalization of the Gauss- Seidel method

by Felix Ogban

2017

In PageRank calculation the Jacobi matrix is given by d T (damping factor times transition matrix), a sparse matrix. The solution of the iteration is x, if the limit exists. The convergence is guaranteed, if the absolute value of the... more

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On the chaotic behavior of non-flat billiards

by Michal Feckan

2016

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Solution of the quantum initial value problem with transparent boundary conditions

by Alejandro Puga Candelas

2016

Many systems of interest at the forefront of technological development, for example, in quantum computation, consist of weakly interacting elements that obey quantum mechanics. A challenge for modern theoretical physics is to develop a... more

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Semiclassical dynamics based on quantum trajectories

by Sophya Garashchuk

2015, Chemical Physics Letters

We present a trajectory-based method that incorporates quantum effects in the context of Hamiltonian dynamics. It is based on propagation of trajectories in the presence of quantum potential within the hydrodynamic formulation of the... more

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Kinetic Theory for Bubbly Flow I: Collisionless case

by Giovanni Russo

2015, Siam Journal on Applied Mathematics

In this paper an efficient and accurate method for simulating the propagation of a localized solution of the Schrödinger equation with a smooth potential near the semiclassical limit is presented. We are interested computing arbitrarily... more

Fig. 1. Case 1. The solid blue curve is a plot of q(t) vs. t where as the red dashed curve shows the deviation of expected position from the classical value by plotting 100((x) — q(t)) vs. t. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) where j is a solution of the Schrodinger equation with an initial condition given by w¢,(x, 0) and || - || denotes the L? norm. Notice that for a fixed ¢ there is an error and nothing that can be done to improve it. In other words, even if the Gaussian beam is computed perfectly an error will be made. Furthermore, the expectation values deduced from w,, can be wrong as well. For example, the expectation of x, (x), given by the Gaussian beam approximation is q(t), the classical value, but

Fig. 2. Numerical results for case 1. The real part of w is computed from (27) using M = 32, At = 10°”, and ¢ = 10°? and plotted at t = 10. The red circle: represent values at the grid points and the blue curve is the Fourier interpolation. The wave function, y is reconstructed from w using (37). This is plotted ir Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Schrédinger equation yields a new partial differential equation whose solution will depend on the Gaussian wave packe parameters. Our approach can be summarized with the following diagram: The w equation has essentially all the stiffness associated with the limit ¢ — 0 removed. In particular, the number of grid points needed to achieve a certain level of accuracy is reduced as ¢ — 0. The ordinary differential equations used to update the GWP parameters are not stiff as ¢ — 0. However, to find y we must apply the inverse of GWPT which requires us to com- pute them very accurately. However, since this only applies to a small set of ordinary differential equations, this requirement does not present a serious computational burden. The computational cost will be carefully discussed in Section 2.8.

Fig. 3. Numerical results for case 1. The real part of the wave function, yw, is is shown at time t = 10. It was reconstructed using w whose real part is plotted in Fig. 2. The red circles represent y for the 32 grid points in € domain. The solution for w, reconstructed from the red points using (37) and (44) (Fourier interpolation), is plotted using blue dashes. The green line represents the fully resolved solution computed by fourth order spitting and 2'° modes, using a time step 0.001. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) In this test case we consider how our approach behaves as ¢ is varied. The same initial condition as in Case 1 are used here. We fix At = 0.01, M = 32, and o = 10. Our “exact” solution uses N = 24 and At = 0.001 for e=10*, N=2"° and

Fig. 4. This is the same as Fig. 3 except the time is t = 20. Most of the error occurs in the tails of the solution. A close-up of the region in the black box i: shown in Fig. 5.

Fig. 5. A close up of Fig. 4 except here the computed values (only two are visible) are marked by the large red circles with a cross. The small blue circle: represent the solution obtained by Fourier interpolation. The solid green line is the “exact” solution. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. A close up of Fig. 5 except here we have included the Gaussian beam approximation which is plotted with a solid black line. It should be noted that the phase error between the Gaussian beam approximation and the exact solution is O(1).

Fig. 7. The relative L error is plotted as function of time for the first test problem. Here ¢ = 10-°, At = 0.01. The dashed line represents the error of the Gaussian beam. The first number in the legend represent the number of modes used for the w-equation, and the second one is the size of the computational domain, o. The “exact” solution used N = 2'® modes with a time step At ~ 0.001.

Fig. 8. The relative L’ error is plotted as function of time for three different values of ¢. The dashed line represents the error of the Gaussian beam. The nev method used At = 0.01, o = 10, and M = 32 for all values of .

Fig. 9. Real part of the wave function at time T = 10 for the non-Gaussian initial condition given by (46). The green line represents the fully resolved calculation while the red circles represent the computed solution on the original M = 32 point grid. Here we have ¢ = 10~? and o = 10. The lower picture is a close up of the boxed region of the upper figure. It also includes the wave function that is reconstructed from the 32 computed points of w using Fourier interpolation. These are plotted as blue points. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

Fig. 10. Relative error as a function of time for non Gaussian initial condition (Case 3). The blue and black curves correspond to the initial condition given by Eq. (46) (solved with M = 32 and o = 10), and Eq. (47) (solved with M = 64 and o = 14), respectively, using the torsional potential. The red curve corresponds to the case using the initial condition given by (46) (M = 32, o = 10), using a harmonic potential. In the latter case, the error remains small after a large time, and there is no need to approximate the initial condition as a superposition of Gaussian beams. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

Fig. 11. Real part of the wave function at time T = } for the “caustic” initial condition given by (46). The green line represents the fully resolved calculation, the blue dashes are the numerical solution that is reconstructed from w and the red circles are the computed values of w. For this simulation ¢ = 10°, M = 64, o = 12, At = 0.01. The final relative L’ error is 3 x 10~°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Numerical Methods for hyperbolic PDE

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