A new approach for admissibility analysis of the direct discontinuous Galerkin method through Hilbert matrices (Q2804378)

The numerical solution of the one-dimensional heat equation \({U_t - U_{xx} = 0}\) for \((x,t) \in \Omega \times (0,T)\) (\(\Omega \in \mathbb{R}\)) with the initial condition \(U(x,0) = U_0(x)\) for \(x \in \Omega\) and periodic boundary conditions by means of a symmetric direct discontinuous Galerkin (DDG) method is studied. One has to find a function \(u\) such that NEWLINE\[NEWLINE \int_{I_j} u_t v \, \mathrm{d}x - ({\widehat u}_x v)_{j+1/2} + ({\widehat u}_x v)_{j-1/2} + \int_{I_j} u_x v_x \, \mathrm{d}x + ([u]{\widehat v}_x)_{j+1/2} + ([u]{\widehat v}_x)_{j-1/2} = 0 NEWLINE\]NEWLINE for all test functions \(v\) and all \(j=1,2, \ldots, N\) holds. Hereby is \(\Omega = \bigcup_{j=1}^N I_j = \bigcup_{j=1}^N [x_{j-1/2},x_{j+1/2}]\); \(u\) and \(v\) are functions of \(L^2(\Omega)\), which are polynomials of degree \(k\) on the cells \(I_j\), \(w_{j\pm1/2}\) denotes the value of the function \(w\) at \(x = x_{j \pm 1/2}\) (\(w = {\widehat u}_x v, [u]{\widehat v}_x)\). The numerical flux is defined by \({\widehat w}_x = \beta_0 [w]/\Delta x + {\bar w}_x + \beta_1 \Delta x [w_ {xx}]\), where \([w]\) and \(\bar w\) denote jumps and averages across the cell interfaces, \(\Delta x = (\Delta x_j + \Delta x_{j+1})/2\), \(\Delta x_j = x_{j+1/2} - x_{j-1/2}\). An important question is how one has to choose the parameters \(\beta_0\) and \(\beta_1\) such that the resulting semidiscrete symmetric DDG method is stable. An admissibility condition for the numerical flux is defined. The fulfillment of this condition ensures the stability of the semidiscrete DDG method. The author gives an inequality containing \(\beta_0\), \(\beta_1\) and the polynomial degree \(k\) which has to be fulfilled such that the corresponding pair (\(\beta_0,\beta_1\)) leads to an admissible numerical flux. For proving this result the admissibility condition is evaluated by using a polynomial basis. Then it is transformed into an eigenvalue problem containing the inverse of a Hilbert matrix. Some new properties of this inverse are shown. The theoretical results are confirmed by the presented numerical experiments.