Structure-Preserving Schemes for Nonlinear Symmetric Hyperbolic and Thermodynamically Compatible Systems of Partial Differential Equations

Abstract This paper aims at developing exactly energy-conservative and structure-preserving finite volume schemes for the discretisation of first-order symmetric-hyperbolic and thermodynamically compatible (SHTC) systems of partial differential equations in continuum physics. Due to their thermodynamic compatibility the class of SHTC systems satisfies an additional conservation law for the total energy and many PDE in this class of equations also satisfy stationary differential constraints (involutions). First, we propose a simple semi-discrete cell-centered HTC finite volume scheme that employs collocated grids and that is compatible with the total energy conservation law, but which does not satisfy the involutions. Second, we develop a fully discrete semi-implicit finite volume scheme that conserves total energy and which can be proven to satisfy also the involution constraints exactly at the discrete level. This method is a vertex-based staggered semi-implicit scheme that preserves the basic vector calculus identities $$\nabla \cdot \nabla \times \textbf{A}= 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∇</mml:mi> <mml:mo>·</mml:mo> <mml:mi>∇</mml:mi> <mml:mo>×</mml:mo> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$\nabla \times \nabla \phi = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∇</mml:mi> <mml:mo>×</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for any vector and scalar field, respectively, exactly at the discrete level and which is also exactly totally energy conservative. The main key ingredient of the proposed implicit scheme is the fact that it uses a discrete version of the symmetric-hyperbolic Godunov-form of the governing PDE system. This leads naturally to sequences of symmetric and positive definite linear algebraic systems to be solved inside an iterative fixed-point method used in each time step. We apply our new schemes to three different SHTC systems. In particular, we consider i) the equations of nonlinear acoustics, ii) the nonlinear Maxwell equations in the absence of charges, and iii) a nonlinear version of the Maxwell-GLM system. We also show some numerical results to provide evidence of the stated properties of the proposed schemes.