Pdf-files of both the slides and the exercises are also provided on these two The lecture gives an introduction to computational physics for students of the. Introduction to Computational Physics by University of Heidelberg University of Heidelberg PDF | Pages | English. This note covers the following topics. physics, computational physics, theoretical physics and physics and computational science in general were still perceived by the majority of.
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C/C++ PROGRAMMING: Software engineering in C, P.A. Darnell, and P.E. Margolis (Springer-Verlag,. New York NY, ). The C++ programming language. A Practical Introduction to Computational Physics and Scientific Computing. AUTHORED BY KONSTANTINOS N. ANAGNOSTOPOULOS. Physics Department. physics, computational physics, theoretical physics and experimental physics are all physics and computational science in general were still perceived by the.
In this section, we will discuss the most common and useful methods for obtaining eigenvalues and eigenvectors in a matrix eigenvalue problem. NA NA Pages. Ancient societies also had to deal with quantifying their knowledge and events. The index used above includes both the spatial and spin indices. We can conclude that this algorithm cannot be adopted in actual numerical solutions of most physics problems.
The above quadrature is commonly referred to as the trapezoid rule, which has an overall accuracy up to O h 2. We can obtain a quadrature with a higher accuracy by working on two slices together. The third-order term vanishes because of cancelation. In order to pair up all the slices, we have to have an even number of slices.
What happens if we have an odd number of slices, or an even number of points in [a, b]? The following program is an implementation of the Simpson rule for calculating an integral. The output of the above program is 1. Note that we have used only nine mesh points to reach such a high accuracy. In some cases, we may not have the integrand given at uniform data points.
The Simpson rule can easily be generalized to accommodate cases with nonuniform data points. We can rewrite the interpolation in Eq. We can, however, develop an adaptive scheme based on either the trapezoid rule or the Simpson rule to make the error in the evaluation of an integral controllable.
Here we demonstrate such a scheme with the Simpson rule and leave the derivation of a corresponding scheme with the trapezoid rule to Exercise 3. First we can carry out S0 and S1. If it is true, we return S1 as the approximation for the integral. The following program is an implementation of the adaptive Simpson rule for the integral above.
PI; 3. In fact, we can assign a random value to a0 and the result will remain the same within the errorbar. The integral showed up while solving a problem in Jackson It is quite intriguing because the integrand has such a complicated dependence on a0 and x and yet the result is so simple and cannot be obtained from any table or symbolic system. Try it and see whether an analytical solution can be found easily.
The adaptive scheme presented above provides a way of evaluating an integral accurately. It is, however, limited to cases with an integrand that is given continuously in the integration region.
For problems that involve an integrand 61 62 Numerical calculus given at discrete points, the strength of the adaptive scheme is reduced because the data must be interpolated and the interpolation may destroy the accuracy in the adaptive scheme.
In determining whether to use an adaptive scheme or not, we should consider how critical and practical the accuracy sought in the evaluation is against the speed of computation and possible errors due to other factors involved. If such a value exists, we call it a root or zero of the equation.
In this section, we will discuss only single-variable problems and leave the discussion of multivariable cases to Chapter 5, after we have gained some basic knowledge of matrix operations.
The bisection method is the most intuitive method, and the idea is very simple. The Newton method This method is based on linear approximation of a smooth function around its root. The above iterative scheme is known as the Newton method. It is also referred to as the Newton—Raphson method in the literature.
This step is repeated toward the root, as illustrated in Fig. The following program is an implementation of the Newton method. For the same function value, a large step is created for the small-slope case, whereas a small step is made for the large-slope case. There is no such mechanism in the bisection method. The disadvantage of the method is that we need two points in order to start the search process.
Other schemes for the multivariable cases are left to later chapters. So all the root-search schemes discussed so far can be generalized here to search for the extremes of a single-variable function. If it is, we accept the update. Assuming that the interaction potential is V r when the two ions are separated by a distance r , the bond length req is the equilibrium distance when V r is at its minimum.
We will force the algorithm to move toward the minimum of V r. We will still obtain the same result as with the secant method used in the earlier example for this simple problem, because there is only one minimum of V x around the point where we start the search. In the example program above, the search process is forced to move along the direction of descending the function g x when looking for a minimum.
This is why this method is known as the steepestdescent method. Like many other optimization schemes, it converges to a local minimum near the starting point. Active research is still looking for better and more reliable schemes. In the last few decades, several advanced methods have been introduced for dealing with function optimization, most noticeably, the simulated annealing scheme, the Monte Carlo method, and the genetic algorithm and programming.
We will discuss some of these more advanced topics later in the book. From systems at the microscopic scale, such as protons and neutrons in nuclei, to those at the astronomical scale, such as galaxies and stars, scattering processes play a crucial role in determining their structures and dynamics.
In general, a many-body process can be viewed as a sum of many simultaneous two-body scattering events if coherent scattering does not happen. In this section, we will apply the computational methods that we have developed in this and the preceding chapter to study the classical scattering of two particles, interacting with each other through a pairwise potential. Most scattering processes with realistic interaction potentials cannot be solved analytically. Therefore, numerical solutions of a scattering problem become extremely valuable if we want to understand the physical process of particle—particle interaction.
We will assume that the interaction potential between the two particles is spherically symmetric. Thus the total angular momentum and energy of the system are conserved during the scattering. In general, a two-particle system with a spherically symmetric interaction can be viewed as a single particle with a reduced mass moving in a central potential that is identical to the interaction potential.
So the motion of a two-particle system with an isotropic interaction is equivalent to the constant-velocity motion of the center of mass plus the relative motion of two particles that is described by an effective particle of mass m in a central potential V r. Cross section of scattering Now we only need to study the scattering process of a particle with a mass m in a central potential V r. A sketch of the process is given in Fig.
We can relate this center-of-mass cross section to the cross section measured in the laboratory through an inverse coordinate transformation of Eq.
Numerical evaluation of the cross section Because the interaction between two particles is described by a spherically symmetric potential, the angular momentum and the total energy of the system are 3. Combining Eq. From Eq. We use the secant method to solve Eq. Then we use the Simpson rule to calculate the integrals in Eq. Finally, we put all these together to obtain the differential cross section of Eq. The following program is an implementation of the scheme outlined above.
The results obtained with the above program for different values of a are shown in Fig. It is clear that when a is increased, the differential cross section becomes closer to that of the Coulomb scattering, as expected. Exercises 3. Discuss the procedure used for dealing with the boundary points.
Show that: The Richardson extrapolation is a recursive scheme that can be used to improve the accuracy of the evaluation of derivatives. Repeat Exercise 3. Show that the contribution of the last slice is also correctly given by the formula in Section 3.
Develop a program that can calculate the integral with a given integrand f x in the region [a, b] by the Simpson with nonuniform data points. Apply the secant method developed in Section 3. Discuss the procedure for dealing with more than one root in a given region. Write a routine that returns the minimum of a single-variable function g x in a given region [a, b]. Use the steepest-descent method to obtain the stable geometric structures of the clusters. Use the molecular potential given in Eq.
Modify the program in Section 3. Discuss the accuracy of the numerical results by comparing them with an available analytical result. In this chapter, we introduce some basic numerical methods for solving ordinary differential equations. Hydrodynamics and magnetohydrodynamics are treated in Chapter 9.
In general, we can classify ordinary differential equations into three major categories: In reality, a problem may involve more than just one of the categories listed above.
A common situation is that we have to separate several variables by introducing multipliers so that the initial-value problem is isolated from the boundaryvalue or eigenvalue problem. We will cover separation of variables in Chapter 7. Here l is the total number of dynamical variables. This equation can be viewed as a special case of Eq. Extending the formalism developed here to multivariable cases is straightforward.
We will illustrate such an extension in Sections 4. Intuitively, Eq. The accuracy of this algorithm is relatively low. To illustrate the relative error in this algorithm, let us use the program for the one-dimensional motion given in Section 1.
Now if we use a time step of 0. The results from a better algorithm with a corrector to be discussed in Section 4. The accuracy of the Euler algorithm is very low. We have points in one period of the motion, which is a typical number of points chosen in most numerical calculations.
If we go on to the second period, the error will increase further. For problems without periodic motion, the results at a later time would be even worse.
We can conclude that this algorithm cannot be adopted in actual numerical solutions of most physics problems. How can we improve the algorithm so that it will become practical? We can formally rewrite Eq. Because we cannot obtain the integral in Eq. The accuracy in the approximation of the integral determines the accuracy of the solution. The Picard method is an adaptive scheme, with each iteration carried out by using the solution from the previous iteration as the input on the right-hand side of Eq.
For example, we can use the solution from the Euler method as the starting point, and then carry out Picard iterations at each time step. In practice, we need to use a numerical quadrature to carry out the integration on the righthand side of the equation.
Can we avoid such tedious iterations by an intelligent guess of the solution? If we apply this system to the one-dimensional motion studied with the Euler method, we obtain a much better result with the 83 Fig.
The following program is the implementation of this simplest predictor—corrector method to such a problem. Furthermore, the improvement can also sustain long runs with more periods involved. Another way to improve an algorithm is by increasing the number of mesh points j in Eq. Thus we can apply a better quadrature to the integral.
Of course, we can always include more points in the integral of Eq. We can make the accuracy even higher by using a better quadrature.
Let us take the simple model of a motorcycle jump over a gap as an illustrating example. The results for three different taking-off angles are plotted in Fig. The maximum range is about m at a taking-off angle of about The most commonly known and widely used Runge—Kutta method is the one with Eqs. We will give the result here and leave the derivation as an exercise to the reader. Even though we are going to examine only one special system, the approach, as shown below, is quite general and suitable for all other problems.
Consider a pendulum consisting of a light rod of length l and a point mass m attached to the lower end. Assume that the 4. This is a reasonable assumption for a pendulum set in a dense medium under a harmonic driving force. As discussed at the beginning of this chapter, we can write the derivatives as variables. In principle, we can use any method discussed so far to solve this equation set. However, considering the accuracy required for long-time behavior, we use the fourth-order Runge—Kutta method here.
As we will show later from the numerical solutions of Eqs.
Note that generalizing an algorithm for the initial-value problem from the singlevariable case to the multivariable case is straightforward. Other algorithms we have discussed can be generalized in exactly the same fashion. In principle, the pendulum problem has three dynamical variables: This is important because a dynamical system cannot be chaotic unless it has three or more dynamical variables.
The following program is an implementation of the fourth-order Runge—Kutta algorithm as applied to the driven pendulum under damping. Under the given condition the system is apparently periodic.
Here points from 10 time steps are shown. The dynamical behavior of the pendulum shown in Fig. We can modify the program developed here to explore the dynamics of the pendulum through the whole parameter space and many important aspects of chaos.
Several interesting features appear in the results shown in Fig. The system at this point of the parameter space is apparently chaotic. The reason why n is even is that the system is moving away from being periodic to being chaotic; period doubling is one of the routes for a dynaimcal system to develop chaos. The chaotic behavior shown in Fig. For example, if we want to solve an initial-value problem that is described by the differential equation given in Eq.
The solution will follow if we adopt one of the algorithms discussed earlier in this chapter. Typical eigenvalue problems are even more complicated, because at least one more parameter, that is, the eigenvalue, is involved in the equation: Let us take the longitudinal vibrations along an elastic rod as an illustrative example here.
For this problem, we can obtain an analytical solution. We will come back to this problem in Chapter 7 when we discuss the solution of a partial differential equation.
Any other types of boundary conditions can be solved in a similar manner. The key here is to make the problem look like an initial-value problem by introducing an adjustable parameter; the solution is then obtained by varying the parameter. We can combine the secant method for the root search and the fourth-order Runge—Kutta method for initial-value problems to solve the above equation set. Ordinary differential equations 1. We plot both the numerical result obtained from the shooting method and the analytical solution in Fig.
Note that the shooting method provides a very accurate solution of the boundary-value problem. It is also a very general method for both the boundary-value and eigenvalue problems. Boundary-value problems with other types of boundary conditions can be solved in a similar manner. When we apply the shooting method to an eigenvalue problem, the parameter to be adjusted is no longer a parameter introduced but the eigenvalue of the problem. We will demonstrate this in Section 4. If all d x , q x , and s x are smooth, we can solve the equation with the shooting method developed in the preceding section.
The following example program is an implementation of the scheme. This is in clear contrast to the general shooting method, in which many more integrations may be needed depending on how fast the solution converges. The Legendre equation, the Bessel equation, and their related equations in physics are examples of the Sturm—Liouville problem.
Our goal here is to construct an accurate algorithm which can integrate the Sturm—Liouville problem, that is, Eq. Before we discuss how to improve the accuracy of this algorithm, let us work on an illustrating example. The solutions of the Legendre equation are the Legendre polynomials Pl x.
The error is vanishingly small considering the possible rounding error and the inaccuracy in the algorithm. This is accidental of course. Note that the procedure adopted in the above example is quite general and it is the shooting method for the eigenvalue problem.
For equations other than the Sturm—Liouville problem, we can follow exactly the same steps. If we want to have higher accuracy in the algorithm for the Sturm—Liouville problem, we can differentiate Eq. If we combine Eqs. When some of the derivatives needed are not easily obtained analytically, we can evaluate them numerically. In order to maintain the high accuracy of the algorithm, we need to use compatible numerical formulas.
The Numerov algorithm is derived from Eq. For more discussion on this issue, see Simos The algorithms discussed here can be applied to initial-value problems as well as to boundary-value or eigenvalue problems. The Numerov algorithm and the algorithm for the Sturm—Liouville problem usually 4. For more numerical examples of the Sturm—Liouville problem and the Numerov algorithm in the related problems, see Pryce and Onodera For example, the energy levels and transport properties of electrons in nanostructures such as quantum wells, dots, and wires are crucial in the development of the next generation of electronic devices.
A sketch of a typical V x is shown in Fig. In order to solve this eigenvalue problem, we can integrate the equation with the Numerov algorithm from left to right or from right to left of the potential region. This is because an exponentially increasing solution is also a possible solution of the equation and can easily enter the numerical integration to destroy the accuracy of the algorithm.
The rule of thumb is to avoid integrating into the exponential regions, that is, to carry out the solutions from both sides and then match them in the well region. Usually the matching is done at one of the turning points, where the energy is equal to the potential energy, such as xl and xr in Fig. The Fig. The turning points x l and x r are also indicated.
This region should be large enough compared with the effective region of the potential to have a negligible effect on the solution. We can start the search with a slightly higher value than the last eigenvalue. We need to make sure that no eigenstate is missed. The potential well, eigenvalue, and eigenfunction shown in Fig. The program below implements this scheme. We outline here a combined numerical scheme that utilizes either the Numerov or the Runge—Kutta method to integrate the equation and a minimization scheme to adjust the solution to the desired accuracy.
We can use the analytical results for a square potential that has the same range and average strength of the given potential as the initial guess. Because the convergence is very fast, the initial guess is not very important.
Note that Ar i Ar , Ai. We can use the steepest-descent scheme introduced in Chapter 3 or other optimization schemes given in Chapter 5. Now let us illustrate the scheme outlined above with an actual example. A sketch of the system is given in Fig. The problem is solved with the Numerov algorithm and an optimization scheme. We use the steepest-descent method introduced in Chapter 3 in the following implementation.
Under this choice of effective mass and permittivity, we have given the energy in the unit of the effective Hartree about The second peak appears at an energy slightly above the barriers.
Study of the symmetric barrier case Pang, shows that the transmissivity can reach at least 0. Note that both B1 and B2 are complex, as are the functions y1 x and y2 x. The Runge—Kutta algorithm in this case is much more accurate, because no approximation for the second point is needed. For more details on the application of the Runge—Kutta algorithm and the optimization method in the scattering problem, see Pang Note that it is not now necessary to have the minimization in the scheme.
However, for a more general potential, for example, a nonlinear potential, a combination of an integration scheme and an optimization scheme becomes necessary. Exercises 4. Implement the two-point predictor—corrector method to study the system. Is there a parameter region in which the system is chaotic? What happens if each particle encounters a random force that is in the Gaussian distribution?
Discuss the options in the selection of the parameters involved. In what parameter region does the orbit become chaotic? Explore the properties of the system under different initial conditions. Can the system ever be chaotic? Discuss the accuracy of the result by comparing it with the solution obtained via the fourth-order Runge—Kutta algorithm and with the exact result.
Discuss the accuracy of the result by comparing it with the exact result. Test the program with the double-barrier potential of Eq. Evaluate l numerically and compare it with the exact result. Is the apparent accuracy of O h 6 in the algorithm delivered? Establish a model and set up the relevant equation set that describes the motion of the disk, including the process of the disk falling onto the table. Write a Ordinary differential equations program that solves the equation set.
Does the solution describe the actual experiment well? Establish a model and set up the relevant equation set that describes the motion of the tippe top, including the process of toppling over. Write a program that solves the equation set. Write a program to study this system. Schemes developed for the matrix problems can be applied to the related problems encountered in ordinary and partial differential equations. For example, an eigenvalue problem given in the form of a partial differential equation can be rewritten as a matrix problem.
A boundary-value problem after discretization is essentially a linear algebra problem. Then we have the potential energy U q1 , q2 ,. We have taken the equilibrium potential energy as the zero point. This equation can be rewritten in a matrix form: Note that it is a homogeneous linear equation set. Another example we illustrate here deals with the problems associated with electrical circuits. Let us take the unbalanced Wheatstone bridge shown in Fig. There is a total of three independent loops.
Each loop results in one of three independent equations: We will see in Section 5. A third example lies the calculation of the electronic structure of a manyelectron system. Three protons are arranged in an equilateral triangle. The two electrons in the system are shared by all three protons.
Assuming that we can describe the system by a simple Hamiltonian H, containing one term for the hopping of an electron Fig. This Hamiltonian is called the Hubbard model when it is used to describe highly correlated electronic systems.
The parameters t and U in the Hamiltonian can be obtained from either a quantum chemistry calculation or experimental measurement. Note that the quantum problem here has a mathematical structure similar to that of the classical problem of molecular vibrations.
The total spin and the z component of the total spin commute with the Hamiltonian; thus they are good quantum numbers. We will consider mainly the problems associated with square matrices in this chapter. A variable array x with elements x1 , x2 ,. Equation 5. Otherwise, the product does not exist. Basic matrix operations are extremely important.
The determinant of a triangular matrix is the product of the diagonal elements. Otherwise, it is a singular matrix. If we interchange any two rows or columns of a matrix, its determinant changes only its sign.
The matrix eigenvalue problem can also be viewed as a linear equation set problem solved iteratively. This is known as the iteration method. In Section 5.
We can 5. This, of course, assumes that matrix S is nonsingular. Readers who are not familiar with these aspects should consult one of the standard textbooks, for example, Lewis The inverse and the determinant of a matrix can also be obtained in such a manner. Similar notation is used for the transformed b as well. In each Numerical methods for matrices step of transformation, we eliminate the elements of a column that are not part of the upper triangle.
The procedure is quite simple. This procedure can be continued with the third, fourth,. Because all the diagonal elements are used in the denominators, the scheme would fail if any of them happened to be zero or a very small quantity.
This is the so-called pivoting procedure. This procedure will not change the solutions of the linear equation set but will put them in a different order. Here we will consider only the partial-pivoting scheme, which searches for the pivoting element the one used for the next division only from the remaining elements of the given column. A full-pivoting scheme searches for the pivoting element from all the remaining elements.
After considering both the computing speed and accuracy, the partial-pivoting scheme seems to be a good compromise. This procedure is continued to complete the elimination and transform the original matrix into an upper-triangular matrix. Physically, we do not need to interchange the rows of the matrix when searching for pivoting elements. When we determine the pivoting element, we also rescale the element from each row by the largest element magnitude of that row in order to have a fair comparison.
This rescaling also reduces some 5. The following method is an implementation of the partial-pivoting Gaussian elimination outlined above. Here index stores pivoting order. If we want to preserve it, we can just duplicate it in the method and work on the new matrix and return it after completing the procedure as done in the Lagrange interpolation programs for the data array in Chapter 2.
For an uppertriangular matrix, the determinant is given by the product of all its diagonal elements. Therefore, we can obtain the determinant of a matrix as soon as it is transformed into an upper-triangular matrix.
Here is an example of obtaining the determinant of a matrix with the partial-pivoting Gaussian elimination. Note that we have used the recorded pivoting order to obtain the correct sign of the determinant.
We have used ki as the row index of the pivoting element from the ith column. If we expand b into a unit matrix in the program for solving a linear equation set, the solution corresponding to each column of the unit matrix forms 5.
A simple example of the LU decomposition of a tridiagonal matrix was introduced in Section 2. Here we examine the general case. So Eq. Note that the inverse of a lower-triangular matrix is still a lowertriangular matrix.
The method performs the LU decomposition with nonzero elements of U and nonzero, nondiagonal elements of L stored in the returned matrix.
The Crout factorization is equivalent to rescaling Ui j by Uii for i j from the Doolittle factorization. In general the elements in L and U can be obtained from comparison of the elements of the product of L and U with the elements of A in Eq. If we implement the LU decomposition following the above recursion, we still need to consider the issue of pivoting, which shows up in the denominator of the expression for Ui j or L i j.
We can manage it in exactly the same way as in the Gaussian elimination. In practice, we can always store both L and U in one matrix, with 5. We can also obtain the inverse of a matrix by a method similar to that used in the matrix inversion through the Gaussian elimination. A few examples of them are given in the exercises for this chapter. However, there is another class of problems that are nonlinear in nature but can be solved iteratively with the linear schemes just developed.
Examples include the solution of a set of nonlinear multivariable equations and a search for the maxima or minima of a multivariable function. In this section, we will show how to extend the matrix methods discussed so far to study nonlinear problems. From the numerical point of view, this is also a problem of searching for the global or a local minimum on a potential energy surface. The multivariable Newton method Nonlinear equations can also be solved with matrix techniques.
In Chapter 3, we introduced the Newton method in obtaining the zeros of a single-variable function. Now we would like to extend our study to the solution of a set of multivariable equations.
The accuracy can be improved if we reduce the tolerance further. Extremes of a multivariable function Knowing the solution of a nonlinear equation set, we can develop numerical schemes to obtain the minima or maxima of a multivariable function. In Chapter 3, we introduced the steepest-descent method in the search of an extreme of a multivariable function.
We can also use the multivariable Newton or secant method introduced above for such a purpose, except that special care is needed to ensure that g x decreases increases during the updates for a minimum maximum of g x.
For example, if we want to obtain a minimum of g x , we can update the position vector x following Eq. This will force the updates always to move in the direction of decreasing g x. This scheme has been very successful in many practical problems.
The reason behind its success is still unclear. For more details on the optimization of a function, see Dennis and Schnabel Much work has been done to develop a better numerical method for optimization of a multivariable function in the last few decades. However, the problem of global optimization of a multivariable function with many local minima is still open and may never be solved.
For some discussions on the problem, see Wenzel and Hamacher , Wales and Scheraga , and Baletto et al. Geometric structures of multicharge clusters We now turn to a physics problem, the stable geometric structure of a multicharge cluster, which is extremely important in the analysis of small clusters of atoms, ions, and molecules.
The function to be optimized is the total interaction potential energy of the system, U r1 , r2 ,. This minimum cannot be a global minimum for large clusters, because no relaxation is allowed and a large cluster can have many local minima.
There are 3n coordinates as independent variables in U r1 , r2 ,. However, because the position of the center of mass and rotation around the center of mass do not change the total potential energy U, we remove the center-of-mass motion and the rotational motion around the center of mass during the optimization. We can achieve this by imposing several restrictions on the cluster.
This removes three degrees of freedom of the cluster. Second, we restrict another particle on an axis, for example, the x axis. This removes two more degrees of freedom of the cluster. Finally, we restrict the third particle in a plane, for example, the x y plane. It is worth pointing out that the structure of NaCl 3 is similar to the structure of H2 O 6 discovered by Liu et al. This might be an indication that water molecules are actually partially charged because of the intermolecular polarization.
The eigenvalues do not have to be different. If two or more eigenstates have the same eigenvalue, they are degenerate. The degeneracy of an eigenvalue is the total number of the corresponding eigenstates.
This trinity outlines the emerging field of computational physics. Topics covered includes: This book covers the following topics: This set of lecture notes serves the scope of presenting to you and train you in an algorithmic approach to problems in the sciences, represented here by the unity of three disciplines,physics, mathematics and informatics. This trinity outlines the emerging field of computational physic.
The purpose of this note is demonstrate to students how computers can enable us to both broaden and deepen our understanding of physics by vastly increasing the range of mathematical calculations which we can conveniently perform.
Currently this section contains no detailed description for the page, will update this page soon. About Us Link to us Contact Us. Free Computational Physics Books. Computational Physics Books This section contains free e-books and guides on Computational Physics, some of the resources in this section can be viewed online and some of them can be downloaded.
Introduction to Computational Physics by University of Heidelberg This note covers the following topics: Author s: University of Heidelberg Pages. Computational Physics by Peter Young This note is intended to be of interest to students in other science and engineering departments as well as physics.
Peter Young NA Pages. This book covers the following topics: This set of lecture notes serves the scope of presenting to you and train you in an algorithmic approach to problems in the sciences, represented here by the unity of three disciplines,physics, mathematics and informatics. This trinity outlines the emerging field of computational physic. The purpose of this note is demonstrate to students how computers can enable us to both broaden and deepen our understanding of physics by vastly increasing the range of mathematical calculations which we can conveniently perform.
Currently this section contains no detailed description for the page, will update this page soon. About Us Link to us Contact Us. Free Computational Physics Books. Computational Physics Books This section contains free e-books and guides on Computational Physics, some of the resources in this section can be viewed online and some of them can be downloaded. Introduction to Computational Physics by University of Heidelberg This note covers the following topics: Author s: University of Heidelberg Pages.
Computational Physics by Peter Young This note is intended to be of interest to students in other science and engineering departments as well as physics. Peter Young NA Pages. Computational Physics Lecture Notes by Morten Hjorth Jensen This set of lecture notes serves the scope of presenting to you and train you in an algorithmic approach to problems in the sciences, represented here by the unity of three disciplines, physics, mathematics and informatics.
Morten Hjorth-Jensen Pages.