A convergence theorem for the vanishing viscosity method and for the Lax–Friedrichs schemes, applied to a nonstrictly hyperbolic and nongenuinely nonlinear system is established. Using the theory of compensated compactness we prove convergence of a subsequence in the strong topology. c © 1999 Elsevier Science B.V. All rights reserved.

We propose a simple, fast sweeping method based on the Lax–Friedrichs monotone numerical Hamiltonian to approximate viscosity solutions of arbitrary static Hamilton–Jacobi equations in any number of spatial dimensions. By using the Lax–Friedrichs numerical Hamiltonian, we can easily obtain the solution at a specific grid point in terms of its neighbors, so that a Gauss–Seidel type nonlinear ite...

6 Friedrichs, K. O., Mathematical Aspects of the Quantum Theory of Fields (New York: Interscience, 1953). 7Friedrichs, K. O., Lectures, Summer Conference on Mathematics and Modern Physical Theories, Boulder (1960). 8 Schwartz, J., Mass Renormalization and Spectral Shifts, N.Y.U. Research Report (1960). 9 Friedrichs, K. O., "Spectral theory of operators in Hilbert space," N.Y.U. lecture notes (1...

The systems of conservation laws have been used to model dynamical phase transitions in, for example, the propagating phase boundaries in solids and the van der Waals uid. When integrating such mixed hyperbolic-elliptic systems the Lax-Friedrichs scheme is known to give the correct solutions selected by a viscosity-capillarity criterion except a spike at the phase boundary which does not go awa...

Four explicit finite difference schemes, including Lax-Friedrichs, Nessyahu-Tadmor, Lax-Wendroff and Lax-Wendroff with a nonlinear filter are applied to solve water hammer equations. The schemes solve the equations in a reservoir-pipe-valve with an instantaneous and gradual closure of the valve boundary. The computational results are compared with those of the method of characteristics (MOC), a...

This paper proposes Friedrichs learning as a novel deep methodology that can learn the weak solutions of PDEs via minmax formulation, which transforms PDE problem into minimax optimization to identify solutions. The name "Friedrichs learning" is for highlighting close relationship between our strategy and theory on symmetric systems PDEs. solution test function in formulation are parameterized ...

We discuss a model of a leaky quantum wire and a family of quantum dots described by Laplacian in L(R) with an attractive singular perturbation supported by a line and a finite number of points. The discrete spectrum is shown to be nonempty, and furthermore, the resonance problem can be explicitly solved in this setting; by Birman-Schwinger method it is reformulated into a Friedrichs-type model.

^Let ii be a simply connected domain in C1 with the area measure dA . Let Pa be the orthogonal projection from L?(Q.,dA) onto the closed subspace of antiholomorphic functions in L2(Í2, dA). The Friedrichs operator Tçi associated to Í2 is the operator from the Bergman space L*(Q.) into L2(fi, dA) defined by Tnf = Paf. In this note, some smoothness conditions on the boundary of Í2 are given such ...