User:HerrHartmuth/sandbox

= Similarity to symmetric tridiagonal matrix = Given a given real tridiagonal, unsymmetic matrix

T = \begin{pmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-1} \\ & & & c_{n-1} & a_n \end{pmatrix} $$ where $$b_i \neq c_i $$.

Assume that the product of off-diagonal entries is strictly positive $$b_i c_i > 0 $$ and define a transformation matrix $$D$$ by

D := \operatorname{diag}(\delta_1, \dots, \delta_n) \quad \text{for} \quad \delta_i := \begin{cases} 1 &, \, i=1 \\ \sqrt{\frac{c_{i-1} \dots c_1}{b_{i-1} \dots b_1}} &, \, i=2,\dots,n \,. \end{cases} $$

The similarity transformation $$J:=D^{-1} T D $$ yields a symmetric tridiagonal matrix $$J$$ by

J:=D^{-1} T D = \begin{pmatrix} a_1 & \sqrt{b_1 c_1} \\ \sqrt{b_1 c_1} & a_2 & \sqrt{b_2 c_2} \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & \sqrt{b_{n-1} c_{n-1}} \\ & & & \sqrt{b_{n-1} c_{n-1}} & a_n \end{pmatrix} \,. $$

Note that $$T$$ and $$J$$ have the same eigenvalues.

=Special Case: Real Tridiagonal= In the case of a tridiagonal structure with real elements the eigenvalues and eigenvectors can be derived explicity as



\begin{align} \lambda_k &= a_0 + 2 \sqrt{a_1 a_{-1}} \cos \left( \frac{\pi k}{n+1} \right) \\ v^k &= \left(     \left( \frac{a_1}{a_{-1}} \right)^{1/2} \sin \left( \frac{1 \pi k}{n+1} \right)   , \ldots ,     \left( \frac{a_1}{a_{-1}} \right)^{n/2} \sin \left( \frac{n \pi k}{n+1} \right)   \right)^T \,. \end{align} $$

= Legendre =

Pointwise Evaluations
As shown before the values at the boundary are given by

P_n(1) = 1 \,, \quad P_n(-1) = \begin{cases} 1 & \text{for} \quad n = 2m \\   -1 & \text{for} \quad n = 2m+1 \,. \end{cases} $$ One can show that for $$ x=0 $$ the values are given by

P_n(0) = \begin{cases} \frac{(-1)^{m}}{4^m} \tbinom{2m}{m} & \text{for} \quad n = 2m \\   0 & \text{for} \quad n = 2m+1 \,. \end{cases} $$

$$ 3 $$

1 $$

= Carrier Gen + Recomb =

Radiative recombination
During radiative recombination, a form of spontaneous emission, a photon is emitted with the wavelength corresponding to the energy released. This effect is the basis of LEDs. Because the photon carries relatively little momentum, radiative recombination is significant only in direct bandgap materials.

When photons are present in the material, they can either be absorbed, generating a pair of free carriers, or they can stimulate a recombination event, resulting in a generated photon with similar properties to the one responsible for the event. Absorption is the active process in photodiodes, solar cells, and other semiconductor photodetectors, while stimulated emission is responsible for laser action in laser diodes.

In thermal equilibrium the radiative recombination $$R_0$$ and thermal generation rate $$G_0$$ equal each other

R_0 = G_0 = B_r n_0 p_0 = B_r n_i^2 $$ where $$B_r$$ is called the radiative capture probability and $$n_i$$ the intrinsic carrier density.

Under steady-state conditions the radiative recombination rate $$r$$ and resulting net recombination rate $$U_r$$ are

r = B_r n p  \,, \quad U_r = r-G_0 = B_r \left( np-n_i^2 \right) $$ where the carrier densities $$n,p$$ are made up of equilibrium $$n_0, p_0$$ and excess densities $$\Delta n, \Delta p$$

n = n_0 + \Delta n \,, \quad p = p_0 + \Delta p \,. $$ The radiative lifetime $$\tau_r$$ is given by

\tau_r = \frac{\Delta n}{U_r} = \frac{1}{B_r \left( n_0 + p_0 + \Delta n \right)} \,. $$

Auger recombination
In Auger recombination the energy is given to a third carrier, which is excited to a higher energy level without moving to another energy band. After the interaction, the third carrier normally loses its excess energy to thermal vibrations. Since this process is a three-particle interaction, it is normally only significant in non-equilibrium conditions when the carrier density is very high. The Auger effect process is not easily produced, because the third particle would have to begin the process in the unstable high-energy state.

The Auger recombination can be calculated from the equation :
 * $$U_{Aug} = \Gamma_n \,n(np-n_i^2) + \Gamma_p \,p(np-n_i^2)$$

In thermal equilibrium the Auger recombination $$R_A$$ and thermal generation rate $$G_0$$ equal each other

R_A = G_0 = C_n n_0^2 p_0 + C_p n_0 p_0^2 $$ where $$C_n,C_p$$ are the Auger capture probabilities.

The non-equilibrium Auger recombination rate $$r_A$$ and resulting net recombination rate $$U_A$$ under steady-state conditions are

r_A = C_n n^2 p + C_p n p^2 \,, \quad U_A = r_A-G_0 = C_n \left( n^2p-n_0^2 p_0 \right) + C_p \left( np^2- n_0 p_0^2 \right) \,. $$

The Auger lifetime $$\tau_A$$ is given by

\tau_A = \frac{\Delta n}{U_A} = \frac{1}{ n^2C_n + 2n_i^2(C_n+C_p) +p^2C_p } \,. $$

Auger recombination in LEDs
The mechanism causing LED efficiency droop was identified in 2007 as Auger recombination, which met with a mixed reaction. In 2013, an experimental study claimed to have identified Auger recombination as the cause of efficiency droop. However, it remains disputed whether the amount of Auger loss found in this study is sufficient to explain the droop. Other frequently quoted evidence against Auger as the main droop causing mechanism is the low-temperature dependence of this mechanism which is opposite to that found for the drop.



$$ $$$$

= MINRES =

In mathematics, the minimal residual method (MINRES) is an iterative method for the numerical solution of a symmetric but possibly indefinite system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Lanczos algorithm is used to find this vector.

Introduction
One tries to solve the following square system of linear equations

Ax=b $$ where $$ x \in \R^n $$ is unknown and $$ A \in \R^{n \times n}\,, b \in \R^n $$ are given.

In the special case of $$ A $$ being symmetric and positive-definite one can use the Conjugate gradient method. For symmetric and possibly indefinite matrices one uses the MINRES method. In the case of unsymmetric and indefinite matrices one needs to fall back to methods such as the GMRES, or Bi-CG. $$ $$

Krylov space basis
The matrix $$ A $$ is symmetric and thus one can apply the Lanczos method to find an orthogonal basis for the Krylov subspace $$ $$

Denote the Euclidean norm of any vector v  by $$\|v\|$$. Denote the (square) system of linear equations to be solved by
 * $$ Ax = b. \, $$

The matrix A is assumed to be invertible of size m-by-m. Furthermore, it is assumed that b is normalized, i.e., that $$\|b\| = 1$$.

The n-th Krylov subspace for this problem is
 * $$ K_n = K_n(A,b) = \operatorname{span} \, \{ b, Ab, A^2b, \ldots, A^{n-1}b \}. \, $$

GMRES approximates the exact solution of $$Ax = b$$ by the vector $$x_n \in K_n $$ that minimizes the Euclidean norm of the residual $$r_n= Ax_n-b$$.

The vectors $$b,Ab,\ldots A^{n-1}b$$ might be close to linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors $$ q_1, q_2, \ldots, q_n \, $$ which form a basis for $$K_n$$. Hence, the vector $$x_n \in K_n $$ can be written as $$x_n = Q_n y_n $$ with $$ y_n \in \mathbb{R}^n $$, where $$ Q_n $$ is the m-by-n matrix formed by $$ q_1,\ldots,q_n $$.

The Arnoldi process also produces an ($$n+1$$)-by-$$n$$ upper Hessenberg matrix $$\tilde{H}_n$$ with
 * $$ AQ_n = Q_{n+1} \tilde{H}_n. \, $$

Because columns of $$Q_n$$ are orthogonal, we have
 * $$ \| Ax_n - b \| = \| \tilde{H}_n y_n - Q_{n+1}^T b \| = \| \tilde{H}_ny_n - \beta e_1 \|, \, $$

where
 * $$ e_1 = (1,0,0,\ldots,0)^T \, $$

is the first vector in the standard basis of $$\mathbb{R}^{n+1} $$, and
 * $$ \beta = \|b-Ax_0\| \, ,$$

$$x_0$$ being the first trial vector (usually zero). Hence, $$x_n$$ can be found by minimizing the Euclidean norm of the residual
 * $$ r_n = \tilde{H}_n y_n - \beta e_1. $$

This is a linear least squares problem of size n.

This yields the GMRES method. On the $$n$$-th iteration: At every iteration, a matrix-vector product $$A q_n $$ must be computed. This costs about $$2m^2 $$ floating-point operations for general dense matrices of size $$m$$, but the cost can decrease to $$O(m)$$ for sparse matrices. In addition to the matrix-vector product, $$O(nm)$$ floating-point operations must be computed at the n -th iteration.
 * 1) calculate $$ q_n $$ with the Arnoldi method;
 * 2) find the $$ y_n $$ which minimizes $$\|r_n\|$$;
 * 3) compute $$ x_n = Q_n y_n $$;
 * 4) repeat if the residual is not yet small enough.

Convergence
The nth iterate minimizes the residual in the Krylov subspace Kn. Since every subspace is contained in the next subspace, the residual does not increase. After m iterations, where m is the size of the matrix A, the Krylov space Km is the whole of Rm and hence the GMRES method arrives at the exact solution. However, the idea is that after a small number of iterations (relative to m), the vector xn is already a good approximation to the exact solution.

This does not happen in general. Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every nonincreasing sequence a1, …, am&minus;1, am = 0, one can find a matrix A such that the ||rn|| = an for all n, where rn is the residual defined above. In particular, it is possible to find a matrix for which the residual stays constant for m &minus; 1 iterations, and only drops to zero at the last iteration.

In practice, though, GMRES often performs well. This can be proven in specific situations. If the symmetric part of A, that is $$(A^T + A)/2$$, is positive definite, then
 * $$ \|r_n\| \leq \left( 1-\frac{\lambda_{\min}^2(1/2(A^T + A))}{ \lambda_{\max}(A^T A)} \right)^{n/2} \|r_0\|, $$

where $$\lambda_{\mathrm{min}}(M)$$ and $$\lambda_{\mathrm{max}}(M)$$ denote the smallest and largest eigenvalue of the matrix $$M$$, respectively.

If A is symmetric and positive definite, then we even have
 * $$ \|r_n\| \leq \left( \frac{\kappa_2(A)^2-1}{\kappa_2(A)^2} \right)^{n/2} \|r_0\|. $$

where $$\kappa_2(A)$$ denotes the condition number of A in the Euclidean norm.

In the general case, where A is not positive definite, we have
 * $$ \frac{\|r_n\|}{\|b\|} \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)|, \, $$

where Pn denotes the set of polynomials of degree at most n with p(0) = 1, V is the matrix appearing in the spectral decomposition of A, and σ(A) is the spectrum of A. Roughly speaking, this says that fast convergence occurs when the eigenvalues of A are clustered away from the origin and A is not too far from normality.

All these inequalities bound only the residuals instead of the actual error, that is, the distance between the current iterate xn and the exact solution.

Extensions of the method
Like other iterative methods, GMRES is usually combined with a preconditioning method in order to speed up convergence.

The cost of the iterations grow as O(n2), where n is the iteration number. Therefore, the method is sometimes restarted after a number, say k, of iterations, with xk as initial guess. The resulting method is called GMRES(k) or Restarted GMRES. This methods suffers from stagnation in convergence as the restarted subspace is often close to the earlier subspace.

The shortcomings of GMRES and restarted GMRES are addressed by the recycling of Krylov subspace in the GCRO type methods such as GCROT and GCRODR. Recycling of Krylov subspaces in GMRES can also speed up convergence when sequences of linear systems need to be solved.

Comparison with other solvers
The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does.

Another class of methods builds on the unsymmetric Lanczos iteration, in particular the BiCG method. These use a three-term recurrence relation, but they do not attain the minimum residual, and hence the residual does not decrease monotonically for these methods. Convergence is not even guaranteed.

The third class is formed by methods like CGS and BiCGSTAB. These also work with a three-term recurrence relation (hence, without optimality) and they can even terminate prematurely without achieving convergence. The idea behind these methods is to choose the generating polynomials of the iteration sequence suitably.

None of these three classes is the best for all matrices; there are always examples in which one class outperforms the other. Therefore, multiple solvers are tried in practice to see which one is the best for a given problem.

Solving the least squares problem
One part of the GMRES method is to find the vector $$y_n$$ which minimizes
 * $$ \| \tilde{H}_n y_n - \beta e_1 \|. \, $$

Note that $$\tilde{H}_n$$ is an (n + 1)-by-n matrix, hence it gives an over-constrained linear system of n+1 equations for n unknowns.

The minimum can be computed using a QR decomposition: find an (n + 1)-by-(n + 1) orthogonal matrix &Omega;n and an (n + 1)-by-n upper triangular matrix $$\tilde{R}_n$$ such that
 * $$ \Omega_n \tilde{H}_n = \tilde{R}_n. $$

The triangular matrix has one more row than it has columns, so its bottom row consists of zero. Hence, it can be decomposed as
 * $$ \tilde{R}_n = \begin{bmatrix} R_n \\ 0 \end{bmatrix}, $$

where $$R_n$$ is an n-by-n (thus square) triangular matrix.

The QR decomposition can be updated cheaply from one iteration to the next, because the Hessenberg matrices differ only by a row of zeros and a column:
 * $$\tilde{H}_{n+1} = \begin{bmatrix} \tilde{H}_n & h_{n+1} \\ 0 & h_{n+2,n+1} \end{bmatrix}, $$

where hn+1 = (h1,n+1, &hellip;, hn+1,n+1)T. This implies that premultiplying the Hessenberg matrix with &Omega;n, augmented with zeroes and a row with multiplicative identity, yields almost a triangular matrix:
 * $$ \begin{bmatrix} \Omega_n & 0 \\ 0 & 1 \end{bmatrix} \tilde{H}_{n+1} = \begin{bmatrix} R_n & r_{n+1} \\ 0 & \rho \\ 0 & \sigma \end{bmatrix} $$

This would be triangular if &sigma; is zero. To remedy this, one needs the Givens rotation
 * $$ G_n = \begin{bmatrix} I_{n} & 0 & 0 \\ 0 & c_n & s_n \\ 0 & -s_n & c_n \end{bmatrix} $$

where
 * $$ c_n = \frac{\rho}{\sqrt{\rho^2+\sigma^2}} \quad\mbox{and}\quad s_n = \frac{\sigma}{\sqrt{\rho^2+\sigma^2}}. $$

With this Givens rotation, we form
 * $$ \Omega_{n+1} = G_n \begin{bmatrix} \Omega_n & 0 \\ 0 & 1 \end{bmatrix}. $$

Indeed,
 * $$ \Omega_{n+1} \tilde{H}_{n+1} = \begin{bmatrix} R_n & r_{n+1} \\ 0 & r_{n+1,n+1} \\ 0 & 0 \end{bmatrix} \quad\text{with}\quad r_{n+1,n+1} = \sqrt{\rho^2+\sigma^2} $$

is a triangular matrix.

Given the QR decomposition, the minimization problem is easily solved by noting that
 * $$ \| \tilde{H}_n y_n - \beta e_1 \| = \| \Omega_n (\tilde{H}_n y_n - \beta e_1) \| = \| \tilde{R}_n y_n - \beta \Omega_n e_1 \|. $$

Denoting the vector $$\beta\Omega_ne_1$$ by
 * $$ \tilde{g}_n = \begin{bmatrix} g_n \\ \gamma_n \end{bmatrix} $$

with gn &isin; Rn and &gamma;n &isin; R, this is
 * $$ \| \tilde{H}_n y_n - \beta e_1 \| = \| \tilde{R}_n y_n - \beta \Omega_n e_1 \| = \left\| \begin{bmatrix} R_n \\ 0 \end{bmatrix} y_n - \begin{bmatrix} g_n \\ \gamma_n \end{bmatrix} \right\|. $$

The vector y that minimizes this expression is given by
 * $$ y_n = R_n^{-1} g_n. $$

Again, the vectors $$g_n$$ are easy to update.

Convergence
The choice of relaxation factor &omega; is not necessarily easy, and depends upon the properties of the coefficient matrix. In 1947, Ostrowski proved that if $$A$$ is symmetric and positive-definite then $$\rho(L_\omega)<1$$ for $$0<\omega<2 $$. Thus, convergence of the iteration process follows, but we are generally interested in faster convergence rather than just convergence.

Convergence Rate
The convergence rate for the SOR method can be analytically derived. One needs to assume the following Then the convergence rate can be expressed as
 * the relaxation parameter is appropriate: $$ \omega \in (0,2) $$
 * Jacobi's iteration matrix $$ C_\text{Jac}:= I-D^{-1}A $$ has only real eigenvalues
 * Jacobi's method is convergent: $$ \mu := \rho(C_\text{Jac}) < 1 $$
 * a unique solution exists: $$ \det A \neq 0 $$.

\rho(C_\omega) = \begin{cases} \frac{1}{4} \left( \omega \mu + \sqrt{\omega^2 \mu^2-4(\omega-1)} \right)^2\,, & 0 < \omega \leq \omega_\text{opt} \\ \omega -1\,, & \omega_\text{opt} < \omega < 2 \end{cases} $$ where the optimal relaxation parameter is given by

\omega_\text{opt} := 1+ \left( \frac{\mu}{1+\sqrt{1-\mu^2}} \right)^2\,. $$

$$ $$



$$

= ILU - Stability = Concerning the stability of the ILU the following theorem was proven by Meijerink an van der Vorst.

Let $$ A $$ be an M-matrix, the (complete) LU decomposition given by $$ A=\hat{L} \hat{U} $$, and the ILU by $$ A=LU-R $$. Then

|L_{ij}| \leq |\hat{L}_{ij}| \quad \forall \; i,j $$ holds. Thus, the ILU is at least as stable as the (complete) LU decomposition.

= ILU - Definition = For a given matrix $$ A \in \R^{n \times n} $$ one defines the graph $$ G(A) $$ as

G(A) := \left\lbrace (i,j) \in \N^2 : A_{ij} \neq 0 \right\rbrace \,, $$ which is used to define the conditions a sparsity patterns $$ S $$ needs to fulfill

S \subset \left\lbrace 1, \dots, n \right\rbrace^2 \,, \quad \left\lbrace (i,i) : 1 \leq i \leq n \right\rbrace \subset S \,, \quad G(A) \subset S  \,. $$

A decomposition of the form $$ A = LU - R $$ which fulfills is called an incomplete LU decomposition (w.r.t. the sparsity pattern $$ S $$).
 * $$ L \in \R^{n \times n} $$ is a lower unitriangular matrix
 * $$ U \in \R^{n \times n} $$ is an upper triangular matrix
 * $$ L,U $$ are zero outside of the sparsity pattern: $$ L_{ij}=U_{ij}=0 \quad \forall \; (i,j) \notin S $$
 * $$ R \in \R^{n \times n} $$ is zero within the sparsity pattern: $$ R_{ij}=0 \quad \forall \; (i,j) \in S $$

The sparsity pattern of L and U is often chosen to be the same as the sparsity pattern of the original matrix A. If the underlying matrix structure can be referenced by pointers instead of copied, the only extra memory required is for the entries of L and U. This preconditioner is called ILU(0).

$$ $$



$$

= CG - Convergence Theorem =

Define a subset of polynomials as

\Pi_k^* := \left\lbrace \ p \in \Pi_k \ : \ p(0)=1 \ \right\rbrace \,, $$ where $$ \Pi_k $$ is the set of polynomials of maximal degree $$ k $$.

Let $$ \left( \mathbf{x}_k \right)_k $$ be the iterative approximations of the exact solution $$ \mathbf{x}_* $$, and define the errors as $$ \mathbf{e}_k := \mathbf{x}_k - \mathbf{x}_* $$. Now, the rate of convergence can be approximated as

\begin{align} \left\| \mathbf{e}_k \right\|_\mathbf{A} &= \min_{p \in \Pi_k^*} \left\| p(\mathbf{A}) \mathbf{e}_0 \right\|_\mathbf{A} \\ &\leq \min_{p \in \Pi_k^*} \,  \max_{ \lambda \in \sigma(\mathbf{A})} | p(\lambda) | \  \left\|  \mathbf{e}_0 \right\|_\mathbf{A} \\ &\leq 2 \left( \frac{ \sqrt{\kappa(\mathbf{A})}-1 }{ \sqrt{\kappa(\mathbf{A})}+1 } \right)^k \ \left\|  \mathbf{e}_0 \right\|_\mathbf{A} \,, \end{align} $$ where $$ \sigma(\mathbf{A}) $$ denotes the spectrum, and $$ \kappa(\mathbf{A}) $$ denotes the condition number.

Note, the important limit when $$ \kappa(\mathbf{A}) $$ tends to $$ \infty $$

\frac{ \sqrt{\kappa(\mathbf{A})}-1 }{ \sqrt{\kappa(\mathbf{A})}+1 } \approx 1 - \frac{2}{\sqrt{\kappa(\mathbf{A})}} \quad \text{for} \quad \kappa(\mathbf{A}) \gg 1 \,. $$ This limit shows a faster convergence rate compared to the iterative methods of Jacobi or Gauss-Seidel which scale as $$ \approx 1 - \frac{2}{\kappa(\mathbf{A})} $$.

$$ $$



$$

= SOR - Symmetric positive definite case =

In case that the system matrix $$ A $$ is of positive definite type one can show convergence.

Let $$ C=C_\omega = I-\left(\frac{1}{\omega} D+L \right)^{-1} A $$ be the iteration matrix. Then, convergence is guarenteed for

\rho(C_\omega) < 1 \quad \Longleftrightarrow \quad \omega \in (0,2) \,. $$

= Jacobi - Symmetric positive definite case =

In case that the system matrix $$ A $$ is of positive definite type one can show convergence.

Let $$ C=C_\omega = I-\omega D^{-1}A $$ be the iteration matrix. Then, convergence is guarenteed for

\rho(C_\omega) < 1 \quad \Longleftrightarrow \quad 0 < \omega < \frac{2}{\lambda_\text{max} (D^{-1}A)} \,, $$ where $$ \lambda_\text{max} $$ is the maximal eigenvalue.

The spectral radius can be minimized for a particular choice of $$ \omega = \omega_\text{opt} $$ as follows

\min_\omega \rho (C_\omega) = \rho (C_{\omega_\text{opt}}) = 1-\frac{2}{\kappa(D^{-1}A)+1} \quad \text{for} \quad \omega_\text{opt} := \frac{2}{\lambda_\text{min}(D^{-1}A)+\lambda_\text{max}(D^{-1}A)} \,, $$ where $$ \kappa $$ is the matrix' condition number.



$$

= Hyperbolic system of partial differential equations =

The following is a system of $$s$$ first order partial differential equations for $$s$$ unknown functions $$ \vec u = (u_1, \ldots, u_s) $$, $$ \vec u =\vec u (\vec x,t)$$, where $$\vec x \in \mathbb{R}^d$$:


 * $$(*) \quad \frac{\partial \vec u}{\partial t}

+ \sum_{j=1}^d \frac{\partial}{\partial x_j} \vec {f^j} (\vec u) = 0, $$

where $$\vec {f^j} \in C^1(\mathbb{R}^s, \mathbb{R}^s), j = 1, \ldots, d$$ are once continuously differentiable functions, nonlinear in general.

Next, for each $$\vec {f^j}$$ a Jacobian matrix $$s \times s$$ is defined


 * $$A^j:=

\begin{pmatrix} \frac{\partial f_1^j}{\partial u_1} & \cdots & \frac{\partial f_1^j}{\partial u_s} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_s^j}{\partial u_1} & \cdots & \frac{\partial f_s^j}{\partial u_s} \end{pmatrix} ,\text{ for }j = 1, \ldots, d.$$

The system $$(*)$$ is hyperbolic if for all $$\alpha_1, \ldots, \alpha_d \in \mathbb{R}$$ the matrix $$A := \alpha_1 A^1 + \cdots + \alpha_d A^d$$ has only real eigenvalues and is diagonalizable.

If the matrix $$A$$ has s distinct real eigenvalues, it follows that it is diagonalizable. In this case the system $$(*)$$ is called strictly hyperbolic.

If the matrix $$A$$ is symmetric, it follows that it is diagonalizable and the eigenvalues are real. In this case the system $$(*)$$ is called symmetric hyperbolic.

Linear system
The case of a linear  hyperbolic system of conservation laws (with constant coefficients in one space dimension) is given by



\begin{align} \quad \frac{\partial \vec u}{\partial t}   + A \frac{\partial \vec u}{\partial x} &= 0 \quad \text{for} \quad (x,t) \in \R \times (0,\infty) \\ \vec{u}(x,0) &= \vec{u}_0(x) \quad \text{for} \quad x \in \R \,, \end{align} $$ where one solves for the unknown function $$ \vec u : \R \times [0,\infty) \rightarrow \R^s $$ and initial data $$ \vec{u}_0 : \R \rightarrow \R^s $$, and $$ A\in\R^{s\times s} $$ are given.

A hyperbolic system is real diagonalizable



\begin{align} A &= R \Lambda R^{-1} \\ \text{with} \quad \Lambda &= \mathrm{diag}(\lambda_1, \dots, \lambda_s) \in \R^{s \times s} \,, \quad R = (\vec r_1, \dots, \vec r_s)\in \R^{s \times s} \,, \quad R^{-1} = (\vec l_1, \dots, \vec l_s)^T \in \R^{s \times s} \,. \end{align} $$ Thus, the conservation law decouples into $$ s $$ independent transport equations

\begin{align} \quad \frac{\partial \vec u}{\partial t}   + A \frac{\partial \vec u}{\partial x} &= 0 \\ \Leftrightarrow \quad \frac{\partial \vec v}{\partial t}    + \Lambda \frac{\partial \vec v}{\partial x} &= 0 \,, \quad \vec v := R^{-1} \vec u \\ \Leftrightarrow \quad \frac{\partial v_i}{\partial t}   + \lambda_i \frac{\partial v_i}{\partial x} &= 0 \quad \forall \, i \,. \end{align} $$ The general solution is

v_i(x,t) = v_i(x-\lambda_i t,0) \,, $$ and in the original variables for given initial data $$ \vec u_0 $$

\vec u(x,t) = R \vec v(x,t) = \sum_{i=1}^s v_i(x-\lambda_i t,0) \vec r_i = $$



$$ $$ $$

= Example: The Laplace operator = The (continuous) Laplace operator in $$ n $$-dimensions is given by $$ \Delta u(x) = \sum_{i=1}^n \partial_i^2 u(x) $$. The discrete Laplace operator $$ \Delta_h u $$ depends on the dimension $$ n $$.

In 1D the Laplace operator is approximated as

\Delta u(x) = u''(x) \approx \frac{u(x-h)-2u(x)+u(x+h)}{h^2 } =: \Delta_h u(x) \,. $$ This approximation is usually expressed via the following stencil

\frac{1}{h^2} \begin{bmatrix} 1 & -2 & 1 \end{bmatrix} \,. $$

The 2D case shows all the characteristics of the more general nD case. Each second partial derivative needs to be approximated similar to the 1D case

\begin{align} \Delta u(x,y) &= u_{xx}(x,y)+u_{yy}(x,y) \\ &\approx \frac{u(x-h,y)-2u(x)+u(x+h,y) }{h^2} + \frac{u(x,y-h) -2u(x) +u(x,y+h)}{h^2} \\ &= \frac{u(x-h,y)+u(x+h,y) -4u(x)+u(x,y-h)+u(x,y+h)}{h^2} \\ &=: \Delta_h u(x) \,, \end{align} $$ which is usually given by the following stencil

\frac{1}{h^2} \begin{bmatrix} & 1 \\   1 & -4 & 1 \\      & 1   \end{bmatrix} \,. $$

Consistency
Consistency of the above mentioned approximation can be shown for highly regular functions, such as $$ u \in C^4(\Omega) $$. The statement is

\Delta u - \Delta_h u = \mathcal{O}(h^2) \,. $$

To proof this one needs to substitute Taylor Series expansions up to order 3 into the discrete Laplace operator.

Subharmonic
Similar to continous subharmonic functions one can define subharmonic functions for finite-difference approximations $$u_h$$

-\Delta_h u_h \leq 0 \,. $$

Mean value
One can define a general stencil of positive type via

\begin{bmatrix} & \alpha_N \\ \alpha_W & -\alpha_C & \alpha_E \\ & \alpha_S \end{bmatrix} \,, \quad \alpha_i >0\,, \quad \alpha_C = \sum_{i\in \{N,E,S,W\}} \alpha_i \,. $$

If $$ u_h $$ is (discrete) subharmonic, then the following mean value property holds

u_h(x_C) \leq \frac{ \sum_{i\in \{N,E,S,W\}} \alpha_i u_h(x_i) }{ \sum_{i\in \{N,E,S,W\}} \alpha_i } \,, $$ where the approximation is evaluated on points of the grid, and the stencil is assumed to be of positive type.

A similar mean value property also holds for the continuous case.

Maximum principle
For a (discrete) subharmonic function $$ u_h $$ the following holds

\max_{\Omega_h} u_h \leq \max_{\partial \Omega_h} u_h \,, $$ where $$ \Omega_h, \partial\Omega_h $$ are discretizations of the continuous domain $$ \Omega $$, respectively the boundary $$ \partial \Omega $$.

= Discontinuous Galerkin Scheme =

Scalar hyperbolic conservation law
A scalar hyperbolic conservation law is of the form

\begin{align} \partial_t u + \partial_x f(u) &= 0 \quad \text{for} \quad t>0,\, x\in \R \\ u(0,x) &= u_0(x)\,, \end{align} $$ where one tries to solve for the unknown scalar function $$ u \equiv u(t,x) $$, and the functions $$ f,u_0 $$ are typically given.

Space discretization
The $$ x $$-space will be discretized as

\R = \bigcup_k I_k \,, \quad I_k := \left( x_k, x_{k+1} \right) \quad \text{for} \quad x_k<x_{k+1}\,. $$ Furthermore, we need the following definitions

h_k := | I_k | \,, \quad h := \sup_k h_k \,, \quad \hat{x}_k := x_k + \frac{h_k}{2}\,. $$

Basis for function space
We derive the basis representation for the function space of our solution $$ u $$. The function space is defined as

S_h^p := \left\lbrace v \in L^2(\R) \; \colon \; {v|}_{I_k} \in \Pi_p \right\rbrace \quad \text{for} \quad p \in \N_0 \,, $$ where $$ {v|}_{I_k} $$ denotes the restriction of $$ v $$ onto the interval $$ I_k $$, and $$ \Pi_p $$ denotes the space of polynomials of maximal degree $$ p $$. The index $$ h $$ should show the relation to an underlying discretization given by $$ \left(x_k\right)_k $$. Note here that $$ v $$ is not uniquely defined at the intersection points $$ \left(x_k\right)_k $$.

At first we make use of a specific polynomial basis on the interval $$ [-1,1] $$, the Legendre_polynomials $$ \left(P_n\right)_{n\in\N_0} $$, i.e.,

P_0(x) = 1 \,,\quad P_1(x)=x \,,\quad P_2(x) = \frac{1}{2}\left( 3x^2-1 \right) \,, \dots $$ Note especially the orthogonality relations

\left\langle P_i,P_j \right\rangle_{L^2([-1,1])} = \frac{2}{2i+1} \delta_{ij} \quad \forall \, i,j \in \N_0 \,. $$ Transformation onto the interval $$ [0,1] $$, and normalization is achieved by functions $$ \left(\phi_i\right)_i $$

\phi_i (x) := \sqrt{2i+1} P_i(2x-1) \quad \text{for} \quad x\in [0,1]\,, $$ which fulfill the orthonormality relation

\left\langle \phi_i,\phi_j \right\rangle_{L^2([0,1])} = \delta_{ij} \quad \forall \, i,j \in \N_0 \,. $$ Transformation onto an interval $$ I_k $$ is given by $$ \left( \bar{\varphi}_{ki}\right)_i $$

\bar{\varphi}_{ki} := \frac{1}{\sqrt{h_k}} \phi_i \left( \frac{x-x_k}{h_k} \right) \quad \text{for} \quad x\in I_k\,, $$ which fulfill

\left\langle \bar{\varphi}_{ki},\bar{\varphi}_{kj} \right\rangle_{L^2(I_k)} = \delta_{ij} \quad \forall \, i,j \in \N_0 \forall \, k \,. $$ For $$ L^\infty $$-normalization we define $$ \varphi_{ki}:= \sqrt{h_k} \bar{\varphi}_{ki} $$, and for $$ L^1 $$-normalization we define $$ \tilde{\varphi}_{ki}:= \frac{1}{\sqrt{h_k}} \bar{\varphi}_{ki} $$, s.t.

\| \varphi_{ki} \|_{L^\infty (I_k) } = \| \phi_i \|_{L^\infty ([0,1]) } =: c_{i,\infty} \quad \text{and} \quad \| \tilde{\varphi}_{ki} \|_{L^1 (I_k) } = \| \phi_i \|_{L^1 ([0,1]) } =: c_{i,1} \,. $$

Finally, we can define the basis representation of our solutions $$ u_h $$

\begin{align} u_h(t,x) :=& \sum_{i=0}^p u_{ki}(t) \varphi_{ki} (x) \quad \text{for} \quad x \in (x_k,x_{k+1}) \\ u_{ki} (t) =& \left\langle u_h(t, \cdot ),\tilde{\varphi}_{ki} \right\rangle_{L^2(I_k)} \,. \end{align} $$ Note here, that $$ u_h $$ is not defined at the interface positions.

DG-Scheme
The conservation law is transformed into its weak form by multiplying with test functions, and integration over test intervals

\begin{align} \partial_t u + \partial_x f(u) &= 0 \\ \Rightarrow \quad \left\langle \partial_t u, v \right\rangle_{L^2(I_k)} + \left\langle \partial_x f(u), v \right\rangle_{L^2(I_k)} &= 0   \quad \text{for} \quad \forall \, v \in S_h^p \\ \Leftrightarrow \quad \left\langle \partial_t u, \tilde{\varphi}_{ki} \right\rangle_{L^2(I_k)} + \left\langle \partial_x f(u), \tilde{\varphi}_{ki} \right\rangle_{L^2(I_k)} &= 0   \quad \text{for} \quad \forall \, k \; \forall\, i \leq p       \,. \end{align} $$ By using partial integration one is left with

\begin{align} \frac{\mathrm d}{\mathrm d t} u_{ki}(t) + f(u(t, x_{k+1} )) \tilde{\varphi}_{ki}(x_{k+1}) - f(u(t, x_k )) \tilde{\varphi}_{ki}(x_k) - \left\langle f(u(t,\,\cdot\,)), \tilde{\varphi}_{ki}' \right\rangle_{L^2(I_k)} =0   \quad \text{for} \quad \forall \, k \; \forall\, i \leq p       \,. \end{align} $$ The fluxes at the interfaces are approximated by numerical fluxes $$g$$ with

g_k := g(u_k^-,u_k^+) \,, \quad u_k^{\pm} := u(t,x_k^{\pm}) \,, $$ where $$u_k^{\pm}$$ denotes the left- and right-hand sided limits. Finally, the DG-Scheme can be written as

\begin{align} \frac{\mathrm d}{\mathrm d t} u_{ki}(t) + g_{k+1} \tilde{\varphi}_{ki}(x_{k+1}) - g_k \tilde{\varphi}_{ki}(x_k) - \left\langle f(u(t,\,\cdot\,)), \tilde{\varphi}_{ki}' \right\rangle_{L^2(I_k)} =0   \quad \text{for} \quad \forall \, k \; \forall\, i \leq p       \,. \end{align} $$