### The a(4) Scheme: A High Order Neutrally Stable CESE Solver

Sin-Chung Chang
2007 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit   unpublished
The CESE development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a non-dissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new high order (4-5th order) and neutrally stable CESE solver of the advection equation ∂u/∂t + a∂u/ ∂x = 0. The space-time
more » ... stencil of this two-level explicit scheme is formed by one point at the upper time level and two points at the lower time level. Because it is associated with four independent mesh variables u^ , (u . , )^ , (u .,. , ) n j ,and (u .,.,. , ) n j (the numerical analogues of u, ∂u/∂x, ∂ 2 u/∂x 2 , and ∂ 3 u/∂x 3 , respectively) and four equations per mesh point, the new scheme is referred to as the a (4) scheme. As in the case of other similar CESE neutrally stable solvers, the a (4) scheme enforces conservation laws in space-time locally and globally, and it has def the basic, forward marching, and backward marching forms. Assuming |ν| 3 these forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove that the a (4) scheme must be neutrally stable when it is stable. Numerically, it has been established that the scheme is stable if |ν| < 1 /3.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8770829,"math_prob":0.93671656,"size":893,"snap":"2023-40-2023-50","text_gpt3_token_len":250,"char_repetition_ratio":0.09448819,"word_repetition_ratio":0.0,"special_character_ratio":0.2362822,"punctuation_ratio":0.09411765,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9707189,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-28T15:52:38Z\",\"WARC-Record-ID\":\"<urn:uuid:1ffb2d28-49ec-4e4e-86c3-cc92e9d2cc72>\",\"Content-Length\":\"17179\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aa23d4b5-d23a-4192-9ae4-ba1fae40b77e>\",\"WARC-Concurrent-To\":\"<urn:uuid:e7cca6cf-1804-45e4-b6da-fd17d600e6d6>\",\"WARC-IP-Address\":\"207.241.225.9\",\"WARC-Target-URI\":\"https://scholar.archive.org/work/fyokpulo2bhrvl5je6ufrrzzwe\",\"WARC-Payload-Digest\":\"sha1:U3WLS5UHM6KB7BJQUVJWVR7ZIKISIFHC\",\"WARC-Block-Digest\":\"sha1:MDJ5L3MKBGSGZO5IT3GYIHP6A2THSAPH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510412.43_warc_CC-MAIN-20230928130936-20230928160936-00707.warc.gz\"}"}